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Lightning surge propagation in overhead lines and gas insulated bus-ducts and cables 1980

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LIGHTNING SURGE PROPAGATION IN OVERHEAD LINES AND GAS INSULATED BUS-DUCTS AND CABLES by |^LEE KAI-CHUNG B.ScT, Uni v e r s i t y of Wisconsin, 1973 M.Sc., Univ e r s i t y of B r i t i s h Columbia, 1975 M.A.Sc., Univ e r s i t y of B r i t i s h Columbia, 1977 r A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY The Faculty of Graduate Studies i n the Department '_ of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA Jul y , 1980 0 Lee Kai-Chung, 1980 In presenting th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of ____________________ The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1WS Date ABSTRACT The propagation c h a r a c t e r i s t i c s of l i g h t n i n g surges i n compressed SFg gas insulated power substation was studied using an electromagnetic transients program. Numerical models were developed to represent the behaviour of d i f f e r e n t system components e s p e c i a l l y under l i g h t n i n g over- voltage conditions. The c h a r a c t e r i s t i c s of l i g h t n i n g surge propagation i n overhead multi-phase untransposed transmission l i n e s was analysed f i r s t . Modal an a l y s i s , :tpgether with s p e c i a l r o t a t i o n techniques to f i t time domain solutions were then used to simulate the wave propagation i n multi-phase untransposed l i n e i n an electromagnetic transients program. Non-linear voltage-dependent corona attenuation and d i s t o r t i o n phenomena were also investigated. Available f i e l d test r e s u l t s could be duplicated to within 5%. The c h a r a c t e r i s t i c s of l i g h t n i n g surge propagation in multi-phase single-core SF^ cables was studied next. A program was developed to obtain the cable parameters for t y p i c a l cable configurations. The amount of core current returning through i t s own sheath and through the earth were computed to i l l u s t r a t e the single phase cable representation for wave propagation i n si n g l e core SFft cables. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i ACKNOWLEDGEMENTS v i INTRODUCTION 1 CHAPTER 1 - LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction 3 2. Lightning discharge mechanism 3 3. S t a t i s t i c a l c h a r a c t e r i s t i c s of l i g h t n i n g strokes 7 4. Frequency of l i g h t n i n g strokes to earth 9 5. Frequency of l i g h t n i n g strokes to power l i n e s . . 12 6. Shielding f a i l u r e phenomenon of l i g h t n i n g strokes 13 CHAPTER 2 - LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduction 18 2. Modal analysis for N-phase untransposed l i n e . . . 19 3. Rotation of eigenvectors f o r zero shunt conductance 22 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine 25 5. Real-valued frequency-independent transformation matrix 25 6. Frequency dependent e f f e c t i n l i g h t n i n g surge propagation 27 7. Determination of surge impedance of the struck phase of a transmission l i n e 33 8. Single phase representation for close-by strokes on double c i r c u i t e d l i n e 36 i i i CHAPTER 3 - LIGHTNING WAVE PROPAGATION IN SF & GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM 1. Introduction 46 2. Formation of series impedance matrix f o r SF, cables 49 6 3. Calcu l a t i o n of s e l f and mutual earth return. . . . 50 4. Calcu l a t i o n of s e l f impedance matrix f o r sin g l e core cable. 61 5. Sheath current return c h a r a c t e r i s t i c s f o r usual earth 67 6. Sheath current return c h a r a c t e r i s t i c s f o r substation earth with grounding g r i d network . . . 74 7. Formation of shunt admittance matrix for SFg cables 79 8. Confirmation of numerical accuracy f o r cable parameter c a l c u l a t i o n and current return r a t i o . . 80 9. Single phase representation parameters f o r multi-phase SF^ cables 80 10. Wave propagation i n SF^ cables 84 CHAPTER 4 - CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES 1. Introduction 87 2. Ph y s i c a l properties of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s 87 3. Transmission l i n e equations for coronated l i n e s . . 90 4. Solution of l i n e equations by compensation method with trapezoidal rules 91 5. Influence on corona by adjacent sub-conductors i n the same bundle 97 6. Influence on corona by adjacent phase conductors . 99 7. Optimal lumping locations and number of corona branch legs 99 8. Overa l l numerical modelling f o r corona e f f e c t s . . 102 i v CHAPTER 5 - CONCLUSIONS 106 APPENDIX A - SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY 107 BIBLIOGRAPHY 109 v ACKNOWLEDGEMENT I would l i k e to show my deepest appreciation to my thesis supervisor Professor Hermann W. Dommel for providing such a unique chance in doing research i n electromagnetic transients of power systems. Dr. Dommel's valuable c r i t i c i s m , and countless hours of discussion during research are also g r a t e f u l l y acknowledged. I am g r a t e f u l to the B.C. Hydro Engineers, Messrs. Jack Sawada, Brent Hughes, Ken Nishikawara and Nick Cuk for t h e i r h e l p f u l discussions. I am also thankful to my fellow graduate students and colleagues Messrs. Obed Abledu, But-Chung Chiu, Shi Wei for t h e i r d i f f e r e n t points of view. The f i n a n c i a l support from the Un i v e r s i t y of B r i t i s h Columbia i n form of teaching and research a s s i s t a n t s h i p and fellowship i s very much appreciated. The f i n a n c i a l assistance of the Systems Engineering D i v i s i o n of the B r i t i s h Columbia Hydro and Power Authority through a Power Systems Research Agreement, and the B r i t i s h Columbia Telephone Company Graduate Scholarship are also g r a t e f u l l y acknowledged. Special thanks are expressed to Miss G a i l Hrehorka i n the E l e c t r i c a l Engineering Main O f f i c e for producing t h i s e x c e l l e n t l y typed t h e s i s . F i n a l l y , I am indebted to my parents for t h e i r continuous encouragement and my wife for her patience. v i 1 INTRODUCTION Every year, atmospheric l i g h t n i n g discharges cause numerous disturbances and damages to e l e c t r i c power systems, such as, destroying transformers and causing black-outs of large areas. This thesis i s devoted to the analysis of l i g h t n i n g surge propagation into compressed SFg gas-insulated substations. The MICA project of the B r i t i s h Columbia Hydro and Power Authority was chosen as a t e s t example. Insulation co-ordination requirements are usually derived from simulated surge propagation studies. This thesis shows that the present p r a c t i c e of i n s u l a t i o n co-ordination design can be improved with the numerical models developed i n t h i s t h e s i s . The contributions of t h i s thesis to i n s u l a t i o n co-ordination design and r e l a t e d power system studies includes the following: 1. Determination of wave propagation i n untransposed l i n e s - Analysis i s used, with a s p e c i a l r o t a t i o n of modal parameters and transformation matrices to make the method sui t a b l e f or time-domain solutions of wave propagation i n multi-phase untransposed l i n e . The s u i t a b i l i t y of d i f f e r e n t s i m p l i f i e d transmission l i n e models i s c l a r i f i e d by comparing simulation r e s u l t s with those from an exact multi-phase rep re sentat ion. 2. Representation of non-linear voltage-dependent corona e f f e c t s - Corona d i s t o r t i o n and attenuation has been simulated with voltage dependent 44 12 v e l o c i t i e s and c o r r e c t i o n factors i n the past ' , or with f i n i t e d ifference methods. However, these methods are i n e f f i c i e n t for d i g i t a l computer a p p l i c a t i o n s . More e f f i c i e n t computational algorithms, using compensation methods, are developed i n t h i s thesis to 2 investigate the non-linear voltage dependent corona e f f e c t s . 3. Determination of wave propagation i n multi-phase si n g l e core SF^ cables - Published methods for the c a l c u l a t i o n of cable constants give inconsistent r e s u l t s . A new cable constants program f o r m u l t i - phase single core SF^-cables has been developed by the author, using various converging.Mnfinite s e r i e s . The complete s h i e l d i n g e f f e c t of the exter n a l l y grounded sheath at frequencies above 1 k Hz has been confirmed with t h i s program. The problem of transient groundrise caused by i n t e r n a l breakdowns or by l i g h t n i n g impulses, as studied by Ontario Hydro"^, i s not included i n t h i s t h e s i s . These transient p o t e n t i a l d i f f e r e n c e s between SF^ bus- ducts and ground occur mainly at the junction with the overhead l i n e . As t h i s thesis shows, the current return i n the SF bus-duct i s completely o through the sheath at frequencies above 1 kHz, whereas the current return of the l i g h t n i n g impulse on the overhead l i n e i s i n the ground. At the junction, the return current must therefore pass from the sheath into the ground through the ground leads, which i n turn causes the transient groundrise problem. These transient groundrises are an important factor i n the design of the grounding system, because they can cause damage to ru a u x i l i a r y wiring or shocks to personnel. 3 CHAPTER 1: LIGHTNING CHARACTERISTICS AND STROKES TO POWER TRANSMISSION LINES 1. Introduction The f i r s t important experiment on l i g h t n i n g was done by Benjamin F r a n k l i n , who used f l y i n g k i t e s to show that l i g h t n i n g i s e l e c t r i c a l i n nature. For more than two centuries, l i g h t n i n g has been the subject of acti v e research. Much of t h i s research has been concerned with the pro- te c t i o n of people and property against the e f f e c t s of l i g h t n i n g stroke. 2. Lightning discharge mechanisms Lightning strokes are f i r s t i n i t i a t e d i n s i d e thunder-clouds. A thunder-cloud usually contains several negative and p o s i t i v e charge centres d i s t r i b u t e d i n d i f f e r e n t locations as shown i n Figure 1.1a. As soon as the e l e c t r o n i s j u m p i n g o v e r t o n e u t r a l i z e the p o s i t i v e charge, a step leader s t a r t s to move down the earth i n d i s c r e t e zig-zag steps of about 50 meters i n length as shown i n Figure 1.1b. This downward p i l o t stroke i s about 1 cm i n diameter and i s not v i s u a l l y detectable by the human eye. As t h i s stepped leader continues to progress downwards, p o s i t i v e charges are induced and accumulated on the ground surface. Eventually, these p o s i t i v e charges jump upwards and form the return stroke to meet the downward stepped leader as shown i n Figure 1.1c. This highly luminous return stroke produces most of the thunder which i s heard. The return stroke i s about 10 cm i n diameter and at a temperature of around 30,000°K. Once a l l the p o s i t i v e charges transfer to the thunder-cloud as shown i n Figure l . l d , the discharged charge centre completely becomes 4 a. Charge n e u t r a l i z a t i o n b. Stepped leader moving within the cloud. downwards. I n i t i a l i z a t i o n of upward d. Complete upward propagation moving return leader. of return leader to cloud (charge centre becomes p o s i t i v e ) . Figure 1.1: Charge d i s t r i b u t i o n and propagation during i n i t i a l l i g h t n i n g discharge. 5 a. Discharge between 2 b. Negative charge dart charge centres. stroke flowing down the continuous earth path. c. Negative charge dart d. Formation of subsequent stroke about to h i t return leader from ground the ground. to cloud charge centres. Figure 1.2: Charge d i s t r i b u t i o n and propagation during sub- sequent dart leader '(multipli-stroke l i g h t n i n g ) . 6 1 p o s i t i v e and s i n g l e stroke l i g h t n i n g discharge i s completed. 2 3 However, about 50% of a l l l i g h t n i n g flashes are multi-strokes ' and contain 3 or 4 subsequent strokes, t y p i c a l l y separated by 30 to 40 ms. About les s than 100 ms a f t e r the f i r s t stroke, a high p o t e n t i a l d i f f e r e n c e i s again established between the charge centres. Discharges again occur and a dart leader i s formed which moves earthwards i n the previous main stream as shown i n Figure 1.2a to 1.2c. S i m i l a r l y , a return stroke i s also formed and more p o s i t i v e charges trans f e r to the thunder-clouds as shown i n Figure 1.2d. The whole process of multi-strokes with r e l a t i v e stroke magnitudes and time scales i s i l l u s t r a t e d i n Figure 1.3a and- 1.3b. T y p i c a l sub- sequent strokes are of the i n i t i a l stroke magnitude and are w e l l separated (about 30 - 40 ms) i n time. The i n i t i a l stroke i s the prime factor i n the i n s u l a t i o n co-ordination studies, but subsequent strokes must be taken into account as a r r e s t e r s must be able to handle r e p e t i t i v e discharges, and the dead times of the auto-reclosing switchgear must be set longer. V e l o c i t y 100% = 300 m/jjs Figure 1.3a: Diagram showing time i n t e r v a l s between i n i t i a l and subsquent strokes.(Ref.3) 100|is lOQuis -35ms- •35ms Figure 1.3b: Current magnitudes of i n i t i a l and subsequent strokes i n t y p i c a l l i g h t n i n g flashes. 3. S t a t i s t i c a l c h a r a c t e r i s t i c s of l i g h t n i n g strokes Due to the d i f f e r e n t d i s t r i b u t i o n s and i n t e n s i t i e s of charge centres ins i d e the thunder-cloud, the c h a r a c t e r i s t i c s of l i g h t n i n g strokes show a large s t a t i s t i c a l v a r i a t i o n i n both magnitude and shape. a. Magnitude of l i g h t n i n g strokes The voltage stress on the power system depends on the magnitude of the l i g h t n i n g current, which i s therefore a c r i t i c a l 4 52 factor i n determining i n s u l a t i o n requirements. -Recorded measurements .' are shown i n Figure 1.4. I t can be seen that 80% of the l i g h t n i n g current magnitudes are within 10 to 100 kA, and only 5% exceed magnitude of 100 kA. 52 It i s suggested that the l i g h t n i n g stroke has to be simulated as an incident current source to the power l i n e with a maximum current magnitude of 100 kA for i n s u l a t i o n co-ordination studies. However, t h i s current source w i l l become an overvoltage wave when propagating down the power l i n e due to the inherent surge impedance of the l i n e . Thus, for 8 Figure 1.4: Cumulative P r o b a b i l i t y of Occurrence of the Amplitudes of Lightning Currents obtained by summarising r e s u l t s from more than 624 measured incidents from 9 countries (Ref. 4). 9 equipment te s t purposes, overvoltage waves are prescribed, b. Waveshape of l i g h t n i n g stroke The l i g h t n i n g waveshapes measured by d i f f e r e n t researchers essen- t i a l l y resemble a double exponential waveshape of d i f f e r e n t r i s e time and decay time. The observed spread of r i s e time i s from very short to 10 us. 5 The observed decay time also spreads from 2 to 100 ps (see Figure 1.5a). The e l e c t r i c power industry therefore agreed many years ago to use a l i g h t n i n g overvoltage wave for equipment i n s u l a t i o n t e s t i n g purpose of a shape 1.2 x 50 ys (explanation-of designation i n Figure 1.5b and 1.5c). Some te s t i n g p r e c r i p t i o n s also specify that t h i s f u l l wave be chopped with a spark gap i n the t a i l to expose the equipment to the higher frequencies which are contained i n the voltage collapse. 4. Frequency of l i g h t n i n g strokes to earth The thunderstorm a c t i v i t y on earth i s measured by the isokeraunic l e v e l . This isokeraunic l e v e l (IKL) gives the number of days per year that thunder has been heard. Usually, thunder cannot be heard outside a 7-24 km radius. An updated world map of isokeraunic l e v e l i s shown i n Figure 1.6. As expected, higher IKL i s found within the t r o p i c a l and sub-tropical regions close to the equator. A f t e r obtaining the IKL of a given place, the number of strokes 2 7 to earth per km (N) i n a p a r t i c u l a r l o c a t i o n i s given by N = A (IKL) stroke/(km 2 - yr) where A = 0.1 to 0.2 10 t^=rise time =time to crest t2=decay time =time to h a l f value 10 100 T i m e ^ s ) a. Wave fronts and t a i l s of l i g h t n i n g surges Time Double exponential wave V = v^ ( e - * t e r ) s wave Time to crest t 1 =1.67(X2-X ) . =1.2jus Time to h a l f value t =X.-X I 4 o =50JJS Time c. T y p i c a l surge waveform impulse generator Figure 1.5: Waveshape of l i g h t n i n g strokes.(Ref.5) Figure 1.6: World D i s t r i b u t i o n of Thunderstorm Days (Ref.6) 12 5. Frequency of l i g h t n i n g strokes to power l i n e s For estimating the number of l i g h t n i n g strokes to power l i n e s , we can start, from the ' e l e c t r i c a l shadow' cast on the ground by the t a l l tower structure with power l i n e s . The frequency of l i g h t n i n g strokes on the ' e l e c t r i c a l shadow' i s assumed to be the frequency of strokes to the power l i n e s . The width (w) of the shadow area estimated by reference 6 i s chosen. For a power l i n e with two ground wires, the width i s given by (see Figure 1.7) w = 4h + b where h = height of ground wire i n m b = separation between ground wires S i m i l a r l y , f o r a power l i n e with only 1 ground wire, the width i s given by w = 4h where h = height of ground wire i n m and for power l i n e s without ground wires, the width i s given by w = 4h + b where h = height of phase wire i n m b = separation between outermost phase wires Thus, the number of strokes/km - yr to the power l i n e (N ) i s Li N L = 0.1 (IKL) stroke /km -yr (1.1) 13 For a t y p i c a l 500 kV tower of the MICA Dam Project, for the l i n e close to the substation, we have AT fr\ i w o r > \ 4 x 37.5 + 18.64 L = ( O - 1 ^ 3 0 ) loOO ( 1 ' 2 ) = 0.5 strokes /km - yr GW GW GW = ground wire M . O : PW PW = phase wire 2h b 2h $ s = s h i e l d i n g angle = 23° h = height of ground wire = 37.5 m b = width between ground wire = 18.64 m Figure 1.7: Lightning stroke ' E l e c t r i c a l shadows' of a t y p i c a l 500 kV transmission l i n e . 6. Shielding f a i l u r e phenomenon of l i g h t n i n g strokes As shown i n Figure 1.7, ground wires are designed f o r s h i e l d i n g of the phase wire from d i r e c t l i g h t n i n g strokes. However, l i g h t n i n g strokes 14 could s t i l l 'sneak' through the ground wire and h i t the phase wire. Such shielding f a i l u r e s have been recorded i n various countries for d i f f e r e n t tower conf igurat ions. Maikopar^ derived a shielding f a i l u r e curve based on observed f i e l d data (see Figure 1.8). However, the graph does not shows the f a c t that shie l d i n g f a i l u r e s occur mainly on lower l i g h t n i n g s t r i k e currents. At higher currents, (e.g. >14.2;kA f o r MICA), the phase wire i s e f f e c t i v e l y shielded from l i g h t n i n g strokes. As seen from Figure 1.1 and 1.2, the p i l o t downward stepped leader from the thundercloud i s formed and propagates earthward f r e e l y regardless of the structure on earth i n i t i a l l y . Later, the return strokes i s formed from a ground object closest to the leader t i p , (ground wire, phase wire, or the ground) arid propagates upward to meet the stepped leader to complete the l i g h t n i n g path. This ground object i s the object which w i l l be struck by the l i g h t n i n g stroke. 8 Brown analysed r e s u l t s from the 120,000 km - yr l i n e i n the Path- finder Project and deduced that the target i s not chosen u n t i l the distance between the stepped leader t i p and the prospective object i s shorter than the s t r i k i n g distance r . This s t r i k i n g distance i s related only to the stroke current as 7 i T 0 • 7 5 r g = 7.1 I m where I = current in kA From t h i s s t r i k i n g distance concept, we can develop the e l e c t r o - geometric model, as shown i n Figure 1.9a. The shielding f a i l u r e of ground- wire at lower current amplitudes can c o r r e c t l y explained by t h i s more ref i n e d method. The degree of exposure of d i f f e r e n t conductors i s 15 SHIELD ANGLE - DEGREES Figure 1.8: P r o b a b i l i t y of Shielding F a i l u r e vs. Shield Angle between Ground Wire and Top Phase Conductor/ represented by drawing exposure arcs of s t r i k i n g distance radius, and centred at each i n d i v i d u a l conductors. The i n i t i a l power frequency voltage of the phase conductor i s ignored as t h i s voltage i s comparatively small to the discharge voltage of the l i g h t n i n g strokes. For l i g h t n i n g currents of 10 kA and 14.2 kA, the corresponding exposure of the phase conductor PW i s shown. It can be seen that the phase conductor exposure to l i g h t n i n g stroke i s decreased with increases i n s t r i k i n g current. For current of amplitudes higher than 14.2 kA, for t h i s tower structure i n MICA project, the phase conductor i s e f f e c t i v e l y shielded by the ground wire and the ground as the exposure arc i s n e g l i g i b l e i n s i z e (See Figure 1.9a). 16 BC=phase wire exposed arc for s h i e l d i n g f a i l u r e SI Figure 1.9a: Electrogeometric model with maximum s t r i k i n g distance of 53.3m. Stroke Current in kA Figure 1.9b: Frequency D i s t r i b u t i o n of Shielding F a i l u r e Stroke Currents in case of Shielding F a i l u r e 17 Brown et a l further investigated t h i s s i t u a t i o n by taking the angular d i s t r i b u t i o n of the l i g h t n i n g stroke g(^) into account and evaluated the phase wire exposed arc for d i f f e r e n t stroke currents as x = r h s i n i i ^ l ^ 2 2 where: g(^) = - cos y (1.4) our The d e t a i l e d a n a l y t i c a l r e s u l t s are shown in Figure 1.9b. For MICA tower of maximum s t r i k i n g distance of 53.3 m (I =14.2kA), the r e s u l t shows that l e s s than 1% of sh i e l d i n g f a i l u r e l i g h t n i n g currents to the phase wire w i l l exceed 14 kA. This agrees well with the geometric i n t e r p r e t a t i o n shown in Figure 1.9a. The l i g h t n i n g stroke usually h i t the ground wire or the tower. In t h i s case, a voltage w i l l b u i l d up ."across the in s u l a t o r because of the po t e n t i a l r i s e on the tower crossarms. If the insula t o r f l a s h over (' backflashover') to the phase conductors, then l i g h t n i n g overvoltage surges w i l l appear on the conductors. 18 CHAPTER 2 : LIGHTNING SURGE PROPAGATION IN OVERHEAD TRANSMISSION LINES 1. Introduct ion Propagation of l i g h t n i n g surges due to d i r e c t strokes, or backflash- overs i n overhead l i n e s influences the choice of i n s u l a t i o n r e q uire- ments. One must know the attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s of the l i n e i n order to f i n d the overvoltages entering the substation where most of the equipment i s concentrated. This section t r i e s to answer the questions whether i t i s possible to represent untransposed overhead l i n e s as equivalent single phase l i n e s for the s t r i c k e n conductor with accuracy, and whether s e l f , p o s i t i v e or zero sequence impedances should be used in such single-phase representations? At f i r s t , f i e l d t e s t s r e s u l t s are duplicated by using a Fourier 9 transformation method. This method not only includes the frequency- dependence of the l i n e parameters, but i t also uses the exact complex, frequency-dependent transformation matrix which requires recomputation at each frequency within the frequency range.typical of l i g h t n i n g surges (e.g. 10 k Hz to 1MHz). This method i s recommended for the simulation of distant strokes where the frequency dependent c h a r a c t e r i s t i c s must be included. For close-by l i g h t n i n g strokes, the above frequency-domain solution can be replaced by a simpler time-domain solution method. This method i s based on modal analysis with frequency-independent parameters and r e a l - valued transformation matrices. The r e s u l t s obtained with the simpler time-domain simulation method agree very well ( < 4% deviation) with the accurate frequency-domain simulation method. After confirming the correctness in the time-domain simulation with the exact N - phase representation of the overhead l i n e f o r close-by l i g h t n i n g strokes, the r e s u l t s obtained are thus compared against s i n g l e - phase approximate representations as presently used. Furthermore, a d d i t i o n a l recommendations are made on how to remove unce r t a i n t i e s i n the choice of surge impedance values of overhead l i n e s . ^ I t i s also found that frequency dependence ef f e c t of nearby l i g h t n i n g stroke can be ignored, Line parameters can be chosen at high frequency e.g. at 1 M Hz, and l i n e 11 12 resistance can be ignored as contradictory to the previous f i n d i n g s . ' 2. Modal analysis for N - phase untransposed l i n e The well known transmission l i n e equations describe the propagation of electromagnetic waves on overhead transmission l i n e s . However, contrary to the s i n g l e phase case, the solution to the N- phase case cannot be obtained e a s i l y since each of the N overhead conductors i s mutually coupled to the other conductors. The following two sets of simultaneous second-order p a r t i a l d i f f e r e n t i a l matrix equations describing the change in voltages and currents along the N - phase l i n e must be solved: dVJ phase" dx n x l r phase r phase " 1 Jnxn L J n x l (2.1) d l dx phase" n x l = r Y P h a s e i r v p h a s e i L Jnxn L J n x l (2.2) where [ z p h a s e ] . [Yphase-| [-,-phase j nxn nxn n x l [V phase. J n x l impedance matrix in phase domain admittance matrix i n phase domain phase current vector phase voltage vector 20 The N coupled differential equations in equations (2.1) and (2.2) can.be transformed into N decoupled equations by replacing phase quan- t i t i e s with modal quantities, [ vphas e ] = [ T ] [ vmod e ] (2.3) [ Iphase ] = [ T _ 3 [Imode] (2.4) and by choosing [T ] and [T ±] in a certain way, as described later. Applying equations (2.3) and (2.4) to equations (2.1) and (2.2) gives demode dx [ T , ] " 1 [ z P h a S e ] [T . ] [ I m ° d e ] (2.5) _ j-̂ modê  |.̂.mode-| (2.6) and d l dx mode = [ I . ] " 1 [ Y P h a S e ] [T ] [ .V n o d e ] (2.7) = j-Ymodê  ̂ o d e ^ (2.8) To find x , and replace T̂ ], we f i r s t differentiate equation (2.1) with respect to *" V phase] with equation (2.2): d l 1 dx d V h a s e l d x 2 ^phase-j j.Yphase^ ^phase^ (2.9) With equation (2.3), this can be written in modal quantities as 2„mode dx = t y " 1 [ Z p h a s e ] [ Y p h a S e ] [T v] 1^°**] (2.10) (2.11) I f [T 1 i s the matrix of eigenvectors of I Z p h a S e ] [ y p h a s e ] , then [ A] v becomes a diagonal matrix, with i t s elements being the eigenvalues of j-zphase^ ^ p h a s e ^ S i m i l a r l y , f o r the current q u a n t i t i e s , we have J2Tmode a L_ L dx 2 = [ T . ] - 1 [ Y P H A S £ ] [ Z p h a S e ] [T.] [ I m ° d e ] (2.12) - UJ [ I m ° d e ] (2.13) where [T\] = matrix of eigenvectors of [Y*\ ] [7? ], with [A] being i d e n t i c a l to that i n equation (2.11). Taking the transpose of the expression for [A] i n equation (2.12) and comparing i t with that f o r [A] i n equation (2.10), while remembering that [ Z p ^ a s e ] and [ y P ^ a s e ] are symmetric, gives: [A] = [?±f [ Z p h a S e ] [ Y P H A S £ ] ( [ T . ] ^ " 1 = I T v ] _ 1 t Z P h a S £ ] [ Y P H A S E ] [ T V ] or [T v] = (-[T.] 1")" 1 (2.14) Thus, only one of the matrices or [T ] i s needed. Using only , rm -i • i . i J i <r rr,niode, . r,,mode, the [T^J-matrix, we can obtain the modal parameters of [Z J and [Y J from equation (2.6) as [ Z m ° d e ] = [7±f [ Z p h a S e ] [T.] (2.15) and from equation (2. 8) as [ Y M ° D E ] = [ I . ] - 1 [ Y P h a S e ] ( [ T . ] 1 1 ) - 1 (2.15a) r mode,-! _ t r v p h a s e , - l r m , or [ Y ] = [T.] [Y ] [T.] (2.16) 13 14 In the computer pugram developed for t h i s modal a n a l y s i s , ' equation (2.16) i s used f o r these two reasons: i t does not require the inverse of [T ] and secondly, the program calculates [Y ] f i r s t anyhow, from which [ Y p k a s e ] i s obtained by inversion. [ Y m o d e ] i s then e a s i l y obtained by taking the r e c i p r o c a l of the diagonal elements of the r i g h t - hand side of equation (2.16). [ Z m ° d e ] i s not calculated from equation (2.15), but i n a simpler way from [ Z m ° d e ] = [A] [ Y m o d e ] - ^ (2.17) that i s , each component i s simply „mode l i Ymode ( 2 < l g ) This i s v a l i d because [A] from equation (2.11) can be rewritten as IA] = [ T ^ " 1 [ Z P h a s e ] [ Y P h a S e ] [ T J . = [ T , ] " 1 [ Z p h a S e ] [T.]. [ T . ] " 1 [ Y P h a S e ] [ T J = [ Z m o d e J . [ Y m ° d e ] (2.19) 3. Rotation of eigenvectors f o r zero shunt conductance It has to be noted that the eigenvectors (columns of [T^] or [T y]) are only determined to within a m u l t i p l i c a t i v e constant. Each eigenvector can, therefore, be m u l t i p l i e d with any non-zero complex s c a l a r , and i t w i l l s t i l l be the correct eigenvector. Since we assume zero phase shunt conductances (corona losses w i l l be discussed l a t e r i n Chapter 4), the modal conductances should also be zero. This can be achieved by multiplying the eigenvectors with a properly chosen constant. Then equation (2.8), which i s defined i n the frequency domain, 23 can be rewritten i n the time domain as follows 8.mode 1 3x = [ c m o d e ] 3 modd v 9t In order tp obtain zero modal conductances, a ro t a t i o n scheme i s used which makes the modal admittance matrix [Y m o c^ e] purely imaginary, mode = { } j [ B m ° d e ] ^ L J r o t a t e L J J L J r o t a t e This r o t a t i o n i s equivalent to d i v i d i n g the i - t h eigenvector ( i - t h column of [T ]) by a factor D^. F i r s t , f i n d the angle 6̂  of Y ^ m o d e , as shown in Figure 2.1. Then 90° - 0. D. = e I (2.20) With a l l Dj^'s forming a diagonal matrix [d], the modified matrix of eigenvectors becomes IT.] = [T.] [D] 1 i J r o t a t e • i J L J (2.21) Then from equation (2.15a), [ Y m ° d e ] r o t a t e = [D] [ T . ] " 1 [ Y P h a S e ] ( [ T . ] 1 ) " 1 [D] (2.22) or [ Y m ° d e ] r o t a t e = [D] [ Y m ° d e ] [D] (2.23) Since a l l matrices i n equation (2.23) are diagonal, equation (2.23) i s simply a to t a t i o n of Y m ° d e by an angle (90° - 6^, which according to „ , , r„mode, , Figure 2.1, makes [Y ^rotate P u r e l y imaginary. 24 After [ Y m ° d e 0 i s found from equation (2.23), and [T.] 1 rotate H ' 1 i rotate from equation (2.21), [ Z m ° d e ] i s calculated from n rotate mode [ Ymode - 1 rotate rotate These modal quantities and transformation matrices obtained are c h a r a c t e r i s t i c s of the p a r t i c u l a r design of the untransposed l i n e . These modal parameters and modal transformation matrices are needed as input for the representation of untransposed d i s t r i b u t e d - parameter l i n e s in the time domain s o l u t i o n , such as i n the UBC version of the E l e c t r o - 13 14 magnetic Transients Program as described i n ' Figure 2.1: Complex Y m o d e before and a f t e r r o t a t i o n 25 4. Confirmation of accuracy of eigenvalue and eigenvector subroutine The UBC Computing Centre l i b r a r y subroutine DCEIGN^^ i s chosen to compute the eigenvalues and engenvectors of the [Y]-[Z] matrix. This double p r e c i s i o n subroutine f i r s t reduces the complex matrix to a Hessenburg matrix H. The subdiagonal elements of H are then forced to converge to 49 zero by the modified LR method. Hence the diagonal elements of H converge to the eigenvalues. The eigenvectors can then be obtained by backward su b s t i t u t i o n . The correctness of the program has been checked by comparing i t s output with published r e s u l t s f o r a doub l e - c i r c u i t line"*"^. Both r e s u l t s of modal attenuations and modal v e l o c i t i e s agree to within three d i g i t s (see Table 2.1). The modal matrices [T ] d i f f e r only s l i g h t l y (see Table 2.2) . Table 2.1 UBC & BPA modal analysis r e s u l t s for a 735 kV l i n e 16 Modal attenuation .15998E6 .18438E6 .18497E6 .18606E6 .18615E6 .18614E6 neper/mile BPA .15998E6 .18437E6 .18497E6 .18605E6 .18614E6 .18614E6 Modal v e l o c i t y UBC .61227E-1 .19050E-2 .18209E-2 .54529E-3 .50169E-3 .47704E-3 miles c/s BPA .612E-1 .191E-2 .182 E-2 .544E-3 .502E-2 .475E-3 5. Real-valued frequency - independent transformation matrix Time domain solutions with the transformation matrix [T_^] become d i f f i c u l t i n theory since [T^] i s complex as well as frequency- dependent. Table 2.2 UBC and BPA modal matrix [T ] r e s u l t s for a 735 kV l i n e . 3412-j.0022 .3948-j.0157 ,4822-j.0 . 3412-j.0022 .3948-j.0157 .4822-j.0 . 3412-j.0022 .3955-J.0157 .4832-j.0 .3412-j.0022 .3955-j.0157 .4832-j.0 .5558+j.0 •3324+j.0230 -.3128+J.0248 .5558+j.0 .3324+j.0230 -.3128+J.0248 •5558+j.0612 .3294+J.0594 -.3153-j.0095 •5558+j.0612 .3294+j.0594 -.3153-j.0095 -.4959-j.0262 .5486-j.0 -.1118+J.0056 -.4959-j.0262 .5486-j.0 -.1118+j.0056 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 -.4959+j.0452 .5453-j.0784 -.1105+j.0215 .1730-j.0017 .4647-j.0247 .5410-j.0 -.1730+J.0017 -. 4647+j.0247 -,5410+j.0 .1730-j.0298 .4681-J.0598 .5469-j.0408 -.1730+J.0298 -.4681+j.0598 -.5469+j.0408 .3209+j.0008 .4804+j.O -.4145+j.0304 -.3209-j.0008 -,4804+j.0 .4145-J.0304 .3209+j.0208 .4659+j.0285 -.4064+J.0048 -.3209-j.0208 -.4659-j.0285 .4064-j.0048 .6827+j.0 -.3550-j.0021 •0812+j.0061 -.6827+j.0 .3550+j.0021 -.0812-j.0061 .6827+j.0737 -.3637-j.0422 .0869+j.0159 -.6827+j.0737 •3637-j.0422 -.0869+j.0159 27 However, the imaginary part of the matrix [T ] i s always small 5%) compared with i t s r e a l part. By taking the r e a l part or the magnitude of the matrix i t s e l f , we obtain modal parameters which-are • s t i l l accurate enough ( ^ 2% deviation). Furthermore, the attenuation caused by corona may be much higher than that caused by the series resistance and for close-by strokes, trans- mission l i n e s should be represented as l o s s l e s s . With the approximations, the frequency dependence of the modal transformation matrix disappears. It i s therefore recommended that the complex matrix be approximated by a real-valued, frequency-independent matrix. This makes simulations much easier f o r two reasons: a) A frequency independent modal matrix does not require recompu- ta t i o n of the modal matrix at each frequency considered within the l i g h t n i n g frequency range, e.g. 10 kHz to 100 kHz. b) A real-valued modal matrix enables d i r e c t transient simulation to be performed i n the time domain. 6. Frequency dependent e f f e c t s i n l i g h t n i n g surge propagation To include frequency dependent effects i n transient overvoltage 17 18 studies i s a complicated topic by i t s e l f . Meyer, Dommel and Marti have investigated the t ime domain methods using convolution i n t e g r a l s and weighting functions. However, the frequency domain solutions can also be Q obtained by the Fourier Transformation methods. Though the frequency domain method i s inadequate to account for the non-linear phenomenon (e.g. corona discharge) and the time domain phenomena (e.g. i n s u l a t o r back — flashover or a r r e s t e r operation), i t i s s u f f i c i e n t for the purpose 28 of studying frequency dependent e f f e c t s on l i g h t n i n g surge propagation i n over- head l i n e s . 9 As discussed i n an e a r l i e r work , the frequency domain solutions includes frequency dependence of l i n e parameters. It also uses the exact complex frequency dependent transformation matrix to be computed at each frequency point, and employs the l i n e a r i n t e r p o l a t i o n technique i n evaluating the Fourier Transformation i n t e g r a l s . The r e s u l t s from a measured f i e l d test 19 20 by Ametani ' of a laboratory generated distant l i g h t n i n g wavefront 83.212 km from the substation was s u c c e s s f u l l y duplicated by the author using the Fourier Transformation method. (See Figure 2.2). Due to the frequency de- pendent e f f e c t of the l i n e parameters, an i n i t i a l r i s e time of 2 ys of the wavefront now increased to about 40 ys as the wave t r a v e l l e d down the l i n e . Thus, the frequency dependent e f f e c t must be included for th§ distant l i g h t n i n g stroke case. The l i g h t n i n g waveshape obtained a f t e r the stroke has t r a v e l l e d from the s t r i k i n g point to the substation can then be interfaced with the time domain solutions as used i n an e l e c t r o - 47 magnetic transients program. For close-by l i g h t n i n g strokes, the r e s u l t i n g waveshapes can again be obtained by the Fourier Transformation integrals, and the simpler time domain methods. For the time domain method,- the multi-phase untransposed l i n e can be f i r s t solved by modal analysis using frequency independent parameters and real-valued transformation matrix (as described in previous s e c t i o n s ) . Then, t h i s multi-phase l i n e i s represented by a single phase l i n e approximation. As shown in Figure 2.3, r e s u l t s obtained by a l l these methods agree quite well (< 4% deviation). The single phase l i n e representation with frequency independent e f f e c t i s v a l i d i n t h i s c l o s e - by stroke case because v a r i a t i o n s among the modal a r r i v a l times at range of l i g h t n i n g frequencies are not apparent i n such short distances (e.g. < 2 km). Voltages(p.u) 3<|> with frequency dependence 3<j> without frequency dependence without frequency dependence ime Figure 2.3: Close-by l i g h t n i n g stroke case solved by frequency and time domain methods. 30 t=0 83.212 km A B • C Output voltage (p.u.) 1.0 0.8 0.6 •0.4 0.2 -0.2 f i e l d measurements 3<j> with frequency dependence Figure 2.2: Numerical simulation of over- voltage taking untransposition and frequency dependence into account.(Ref. 9) 31 In such a representation, s e l f impedance - parameters calculated at higher frequencies (e.g. 1 MHz) should be used to approximate the frequency dependance c h a r a c t e r i s t i c s of the l i n e . However, caution must be taken in choosing l i n e r esistance for the l i g h t n i n g surge studies. The frequency dependence of l i n e parameters of one phase for a t y p i c a l 500 kV l i n e i s shown i n Table 2.3. I t i s shown that the attenuation of the wave i s n e g l i g i b l e (< 5%) up to about 100 kHz for 1 km. The resistance to reactance r a t i o i s also small e s p e c i a l l y at higher frequencies, e.g. 2.8% at 1 M Hz. Furthermore, since the Bergeron's method of c h a r a c t e r i s t i c i n solving the transmission l i n e equation i s v a l i d only for a l o s s l e s s transmission l i n e , d i s t r i b u t e d l i n e losses are usually approximated by lumping the resistance at c e r t a i n l o c a t i o n s . This high resistance at 1 M Hz may cause inaccuracy i n the simulation. On the other hand, surge impedances calculated by zsurge = A + J^L ( 2 > 2 5 ) where R/jwL = 2.8% at 1 M Hz (See Table 2.3) zsurge = fcjL ( 2 > 2 6 ) / JUJC are e s s e n t i a l l y i d e n t i c a l for t h i s lossy and l o s s l e s s cases. Thus, com- p l i c a t e d frequency dependent e f f e c t s for the nearby stroke case can be ignored, and . frequency independent and l o s s l e s s representation give acceptable accuracy(.,,See Figure 2.9 ). ./ . Therefore,.the previous methods of modelling l i n e losses by simple • -i j • 1 2 • i 11,21 exponential decay i n overvoltages or any resistance lumping scheme are not acceptable. They should be replaced by d e t a i l e d weighting function techniques, or Fourier Transformation methods for distant stroke, or by 32 Frequency Resistance Reactance R/X (Hz) R(fi/km) X(ft/km) % zsurge v e l o c i t y Attenuation ( a ) (m/ys) e" Y^( /km) i n 6 183. 6525. 2.8 291. 280. .73 i o 5 42. 692. 6. 300. 272. .93 i o 4 7. 78. 9. 317. 257. .989 i o 3 .9 8.9 10. 340. 240. .999 Table 2.3: Frequency dependence of s e l f quantities of l i n e parameters for a 3 phase 500 kV l i n e . 33 l o s s l e s s l i n e representation f o r a nearby stroke as described i n above, 7. Determination of the surge impedance of the struck phase of a transmission l i n e . An accurate and r e l i a b l e value of the surge impedance in phase domain must be obtained as due to the following reasons: a) The amount of overvoltage wave transmitted from the overhead l i n e to the underground SF^ cable at the c a b l e - l i n e junction i s determined by the surge impedances of d i f f e r e n t components. The r e f r a c t i o n c o e f f i c i e n t t C _ i s R cable" C R , 2 Zsurge 7 ) zl:i.ne + ^cable surge surge where ^cable _ g u e i m p e d a n c e of cable (e.g. 60 ft) surge z l i n e _ g u r impedance of l i n e (e.g. 304 ft) surge b) The exact value of the overvoltage wave on the li n e , resulting from the l i g h t n i n g stroke (1^) i s d i r e c t l y r e l a t e d to surge -i . r,line impedance of the lxne Z as r surge v = ^ . z l i n e (2.28) 2 surge This r e s u l t i n g overvoltage wave impresses e l e c t r i c a l stress on external and i n t e r n a l i n s u l a t i o n of the system and forms the main concern i n the i n s u l a t i o n co-ordination study. 34 im In s p i t e of the above important c r i t e r i a , u n c e r t a i n t i e s i n surge pedance ca l c u l a t i o n s of overhead l i n e do e x i s t . ̂ ' ^ ' ^ Reference 11 give r e l a t i v e l y lower surge impedance r e s u l t s f o r the ground wire (352 ft) 12 as compared to Darveniza's computation. Darveniza claims that the equation for surge impedance i n phase domain as i s given by: Z S U * f = 60 In ^ " (2.29) s e l f r Z S U r g € \ = 60 In ^- (2.30) mutual b.. where h = conductor height ; r = conductor radius a. .. = separation between conductors i j b. . = separation between conductor and i j other conductor image This i s r e a d i l y derived from the p o t e n t i a l c o e f f i c i e n t P and the inductance term L as: 5 s e i f • h r r ^ r - " l n T < 2- 3 I> J s e l f L 1 = Ho- to ^h ( 2 - 3 2 ) s e l f 2TT r and Z ^ f = / ^ - - - A _ _ P = 60 £ n ^ (2.33) s e l f / C ... s e l f s e l f ~ s e l f 35 where u = permeability ° -7 = 4TT x 10 H/m e = permit i v i t y ° 1 -9 = x 10 F/m 36TT [C] = [ P ] _ 1 [P] = p o t e n t i a l c o e f f i c i e n t matrix, with diagonal term P s e l f [L] = inductance matrix, with diagonal term L ° s e l f However, the above formulae neglect • Carson's correction terms, other conductors, and ground wires used f o r earth return. A d e t a i l e d c a l c u l a t i o n for the surge impedance matrix i n phase domain [Z ] & r v surge must be performed to in order to j u s t i f y t h i s assumption. If we consider the r e l a t i o n s h i p between the surge impedance matrix in both phase and modal domain as [ y p h a s e ] = [ z p h a s e ] [ ; I p h a s e ] ( 2 _ 3 4 ) and [ V m ° d e ] = [ Z m ° d e ][ I m ° d e ] (2.35) surge then by s u b s t i t u t i n g eqs.(2.3) & (2.4) into (2.35), we can get [T TV1**36] = [ Z m ° d e ][T ] _ 1 [ l p h a S e ] (2.36) v surge I or [ V p h a S e ] = [T ] [ Z m ° d e ] [ T . ] - 1 [ l p h a S e ] (2.37) v surge I 36 Comparing equations (34) and (37) , we thus obtain [Z phase surge ] = [T ][Z .mode 'surge J[T.] -1 (2.38) The above r e l a t i o n i n equation (2.38) i s i d e n t i c a l to that derived by Wedepohl. 22 In h i s method, the r e f l e c t i o n c o e f f i c i e n t f o r phase current i s f i r s t obtained. The c o e f f i c i e n t i s then set to zero to obtain the expression f o r [ Z P ^ a s e ] as in equation (2.38). surge M Results for the surge impedance from equations (2.29) and (2.38) for both the ground and the phase wires are shown in Table 2.4. As can be seen from the table , the surge impedance obtained by Darveniza's formula which neglects the skin e f f e c t of the earth return component introduces n e g l i g i b l e deviation (about 1%). However, the Darveniza's formula should only be used when the ground wire i s treated as another i n d i v i d u a l phase (e.g. for the close-by l i g h t n i n g stroke case). If one takes the ground wire as another component for earth return (e.g. for the distant stroke case), the formula for s e l f surge impedance must be modified accordingly by t r e a t i n g voltages on ground wire to be zero. This requires reducing the impedance and admittance matrices before surge impedances can be calculated. The surge impedance value obtained i n t h i s case i s lower than that obtained by Equation (2.29), as shown in Table 2.4. 8. Single phase representation for close-by strokes on double c i r c u i t e d l i n e A f t e r the author has v e r i f i e d that s i n g l e phase representation with appropriate choice of l i n e parameters i s accurate for a three phase l i n e case without ground wire r a double-circuited overhead transmission l i n e 23 of the MICA Project of the B.C. Hydro and Power Authority was used as a more de t a i l e d transmission system with ground wires. 37 surge Impedances ground wire phase wire A 547 ft 342 ft B or C 545 ft 338 ft D - 318 ft (A-B)/A*100% 0.4% 1.2% A = Exact method (2.38) with Carson.':s Correction terms for earth return skin e f f e c t , ground wire treated as another phase. B = Exact method equation (2.38) without Carson's Correction terms f o r earth return skin e f f e c t . C = Darveniza approximate equation (2.29). D = Exact method equation (2.38) with Carson's Correction terms for earth return skin e f f e c t , ground wire treated as earth return component. Table 2.4: Self surge impedances for ground and ^ 7 phase wires f o r a t y p i c a l 500 kV l i n e ' 38 This tranmission system i s a double-circuited 500 kV l i n e . Each tower consists of a three phase l i n e with two ground wires (See Figures 2 .4 and 2.5). When the l i g h t n i n g stroke h i t s e i t h e r one of the ground wires or one of the phase conductors, d i f f e r e n t l i n e parameters must be chosen because of d i f f e r e n t l i n e design. The corresponding parameters are shown in Table 2.5. One can see from t h i s table that the s e l f surge impedance of the ground wire i s greater than that of the phase conductor. The wave propagation v e l o c i t y i s also lower i n the ground wire case. In Figures 2.6 and 2.7, one can compare the l i g h t n i n g overvoltage wave propagation c h a r a c t e r i s t i c s f or the open and short c i r c u i t t e s t by using m u l t i - and single-phase l i n e representation when l i g h t n i n g stroke, h i t s the ground wire. Figure 2.6a shows the r e s u l t obtained by the m u l t i - phase s o l u t i o n method using modal a n a l y s i s . I t also shows c l e a r l y the d i f f e r e n t modal components on the r e s u l t i n g waveform. Figure 2.6b shows the r e s u l t f o r the si n g l e phase case and the o v e r a l l important propagation c h a r a c t e r t i s t i c s of multi-phase representation i s s u c c e s s f u l l y duplicated here. Similar r e s u l t s are obtained for the s h o r t - c i r c u i t t e s t , as shown i n Figures 2.7a and 2.7b. One can observe that the current waveforms obtained from these two d i f f e r e n t l i n e representations agree very w e l l . S i m i l a r l y , the open and short c i r c u i t t e s t r e s u l t s are also s u c c e s s f u l l y duplicated for surges on the phase conductor as i n c a s e of d i r e c t s t r o k e s or b a c k f l a s h o v e r s ( See Fig.2.8&2.9) Thus, i t i s recommended to use si n g l e phase representation for double-circuited l i n e with ground wires for studying close-by l i g h t n i n g stroke propagations. 23 unit 5-3/4x10" shielding ground tower insulator //wires extending strings / / 1.6km beyond substn 3 phase conduc tors tower Ty p i c a l l i g h t n i n g arrester c h a r a c t e r i s t i c s : nominal rating (reseal voltage), switching sparkover Min 60 Hz sparkover lightning sparkover 3 phase underground SF- cables 6 - l i g h t n i n g a r r e s t e r Jat transformer ^jO^L-^ cable junction .transformer to be protected 420 kV 950 kV 568 kV 985 kV 396 kV 960 kV- 555 kV 990 kV- Insulation levels 60 Hz BIL SF, cable Transformer 6 800 kV 745 kV 1550 kV 1675 kV Figure 2.4: Layout of SFg substation protection scheme showing one of the double c i r c u i t systems. 40 10 — 9 4 1 5 8 7 6 1 2 3 Conductors 1-3, 6-8 phase wires 4,5, 9,10 ground wires Coupling f a c t o r : l i g h t n i n g struck <j) - wire K 34 ^phase surge 3,4 ^phase surge 3,3 •16 y induced ground wire V phase wire l i g h t n i n g struck g - wire K ^phase _ surge 4,3 _ 43 phase Jsurge 4,4 •06 induced phase wire V ground wire Figure 2.5: Side view of the MICA 10 <j> Systems. Ground conductor Phase conductor Self surge impedance Z ^ | e 658 ft 304 ft Wave v e l o c i t y v 245 ™/\is 293 m / y s Line resistance 0 /̂m 0 /̂m Length 1609 m 1609 m where Z S ^ f = 60 In ^ s e l f r = /L P 77 s e l f s e l f v e l o c i t y = s e l f L s e l f and L . , _. and P . , , are diagonal elements of s e l f S 21 matrix [L] and [P]. Table 2.4: Line parameters of ground and phase conductor for l o s s l e s s single phase representation. 42 Dif f e r e n t modal a r r i v a l times Single phase a r r i v a l In 1<|> case: V T 2 p 3 = (Coupling factor) surge = 3 ' 4 V surge T2G4 Z4,4 Figure 2.6: Open c i r c u i t t e s t on s i n g l e and multi-phase representation with stroke on ground wire. 4 3 i D i f f e r e n t modal a r r i v a l times Figure 2.7: Short c i r c u i t t e s t on s i n g l e and m u l t i - phase representation with stroke on ground wire. Figure 2.8: Open c i r c u i t t e s t on sin g l e and multi-phase representation with stroke on phase conductors. 45 Small d i f f e r e n c e i n d i f f e r e n t modal a r r i v a l times _3 i T 2 p 3 ( 1 0 P-u.) t-0- \. 304« TP3 T2P3 1+ 4 2 2 p.u. (S T2P3 Time t} ii,.or 4 f 8 12- 16 (ys) Single phase a r r i v a l Figure 2.9: Short c i r c u i t t e s t on si n g l e and multi-phase representation with stroke on phase conductors. 46 CHAPTER 3: LIGHTNING WAVE PROPAGATION IN SF 6 GAS INSULATED UNDERGROUND TRANSMISSION CABLE SYSTEM. 1. Introduction The world's f i r s t commercial SF, gas - i n s u l a t e d cable rated at 6 345 kV was i n s t a l l e d i n 1970. I t s inherent advantages over conventional underground o i l - f i l l e d cables with respect to charging current, d i e l e c t r i c losses, thermal performance, voltage r a t i n g f l e x i b i l i t y and power handling capacity are well-known. It o f f e r s a d d i t i o n a l advantages of reduced substation s i z e . This compactness i n si z e of SF^-insulated substations and switchgear brings the equipment closer to the protective l i g h t n i n g arrester located at the overhead l i n e and underground cable junction. This i s of.vital-importance, e s p e c i a l l y when there i s no l i g h t n i n g a rrester at the transformer terminal, as in c e r t a i n substation design. 23 In the SF,-insulated cable at the MICA Dam, which w i l l be used o as a test example, each of the 3 phase cables consists of two concentric aluminum tubes (see Figure 3.1a). The inner tube i s the conductor core and the outer grounded tube i s the sheath. The three sheaths are s o l i d l y bonded together and grounded at many lo c a t i o n s . At the high frequencies encountered i n l i g h t n i n g surges, the sheath return current w i l l be equal i n magnitude and 180° out of phase with the core conductor current. Whether the magnetic f i e l d external to the sheaths can be completely neglected in the frequency range of in t e r e s t must be investigated, however. I f the magnetic f i e l d i s n e g l i g i b l e , then there would be no mutual inductive coupling among the phases. There i s no e l e c t r o s t a t i c c apacitive coupling between phases as the solidly-grounded sheaths act as e l e c t r o s t a t i c shields U p to 1 Mhz. 47 Skin depth o f A l , •t * \-1/2 8.1 = ( . T T f a u ) = cm =1.0 cm at 60Hz Scale--= 1:2.54 r = r„ = 3" = 7.62 cm 3.5" = 8.89 cm 9.75" = 24.765 cm r. = 10" = 25.4 cm 4 sheath thickness=.635 cm Permeability A l = y u, = u- r 0 0 = 4TT .x 10 -7 H/ m Figure 3.1a: Individual cable design P e r m i t t i v i t y SF & = £ r £ Q = eQ 1 x 10 9 F/m 36TT Figure 3.1b: Overall cable layout. 48 In order to investigate the sheath current return phenomenon of SF, b cables, and thus to c l a r i f y the wave propagation c h a r a c t e r i s t i c s , the cable parameter must be calculated accurately. In t h i s research work, cable 24 25 parameters of multi-core cable ' i s not investigated. Such multi-core cable systems c e r t a i n l y have coupling between phases at a l l frequencies. Nevertheless, coupling between phases for the s i n g l e core cable system i n d i f f e r e n t frequencies needs further i n v e s t i g a t i o n . 26 27 Commellini and Abledu used f i n i t e elements technique to sub- divide the main conductors into smaller sub-conductors of c y l i n d r i c a l shape. The impedance matrix f o r the main conductors was formed by bundling up the sub-conductors i n the matrix elimination process. However, due to the t h i n tubular shape of the conductors involved i n the SF^ buses, large number of sub-conductors i s required. This w i l l demand huge computer core storage space and long computer execution time. 28 29 Sunde and Pollaczek had derived a n a l y t i c a l expressions for the s e l f and mutual impedance of cables which are constructed overhead, underground or on ground surface. These a n a l y t i c a l expressions contain Kelvin functions and an i n f i n i t e i n t e g r a l known as the Carson's Correction terms. Before the widespread a p p l i c a t i o n of d i g i t a l computers, s i m p l i f i e d assumptions and r e s t r i c t i o n s were made to f a c i l i t a t e the computation process. With the recent popularity and increased a p p l i c a t i o n of d i g i t a l computers, these i n f i n i t e i n t e g r a l s can be modified and replaced by s t r a i g h t forward numerical computations without s i g n i f i c a n t s a c r i f i c e for accuracy. 30 31 32 Wedepohl et a l and Ametani ' used d i f f e r e n t approaches to tackle the a n a l y t i c a l expression i n the cable parameter c a l c u l a t i o n . However, both approaches gave d i f f e r e n t r e s u l t s (20% from each other). Bianchi proposed to ca l c u l a t e the earth or sea return impedance by approximating the return medium as a tube of i n f i n i t e outside radius. These approximation r e s u l t s f e l l somewhere between those of Wedepohl et a l and Ametani. Because of the inconsistency i n the above f i n d i n g s , a d e t a i l e d i n v e s t i g a t i o n f o r numerical c a l c u l a t i o n of cable parameters must be performed to reveal wave propagation c h a r a c t e r i s t i c s i n SF^ sing l e core cables. Current return c h a r a c t e r i s t i c s from core through sheath also must be investigated to confirm s i n g l e phase or multi-phase representation for the cable system involved. 2. Formation of series impedance matrix f o r SFfi cables The s i n g l e core SF^ cable system configuration i s shown in Figure 3. Each phase consists of two conductors, core and sheath. We can b u i l d up a 6 x 6 series impedance matrix Z, describing the cable system as follows: d V c l n dx z i i s l l Z 19 sl2 Zml2 Zml2 Zml3 Zml3 " ^ l " dV . s i dx Z 01 s21 Zs22 Zml2 Zml2 Zml3 Zml3 ^ 1 d V c 2 dx Zml2 Zml2 z i i s l l Z s l 2 3nl2 Zml2 ^ 2 dV s2 dx Zml2 Zml2 Zs21 Zs22 Zml2 Zml2^ • ^ 2 dV _ c3 dx Zml3 Zml3 Zml2 Zml2 Z s l l Z s l 2 ^ 3 dV _ s3 dx _ Zml3 Zml3 Zml2 Zml2 Zs21 Zs22 _ . ^ 3 . 50 z s Zml2 Z, " ml3 ^ 1 Zml2 Z s Zml2 \l ^ 2 (3.1) Zml3' T Zml2 Z s ^3 s where Z is... 'self submatrix on the diagonal. A l l Z matrices are s ° s equal because they represent i d e n t i c a l cable configurations. I t i s assumed: that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are a l l equal. In other words, a l l elements i n the sub-matrix Z , „ or Z. are ^assumed ml2 ml3 to,; be equall:J:-(Bee Section 6 f o r "further discuss ion) • — 3. C a l c u l a t i o n of s e l f and mutual earth return impedance f o r s i n g l e core cable The a n a l y t i c a l expressions f o r the s e l f and mutual earth return 29 impedance of cables was f i r s t derived by Sunde and Pollaczek^ and then by Wedepohl et a£? F i r s t l y , t h e Maxwell's electromagnetic equation can be solved a f t e r neglecting end e f f e c t s as: 51 V x E = -ja>y0 H (3.2) V x H = J + |2. = j ( i + (3.3) 3t a = J , as displacement current , can be neglected. Taking the c u r l of equation (3.2) and su b s t i t u t i n g into (3 3) give V x V x E = -jwy 0 V x H or V ( V E ) - V 2E = - j u y Q ( J ) Assuming cable separation >> cable radius, we have V 2E = jtoy 0 a(E +p i 6(x) <5(y + h)) Assuming cables are p a r a l l e l to ground surface and attenuation of voltage and current i s n e g l i g i b l e over distances comparable to cable separation, 2 2 we have 3 E n 3 E, + —T- = 0 » y > o. ( 3 - 4 > 2 2 3x 3y 2 2 3 E„ 3 E —j- + Y = m 2E 2 +p m2 i 6(x) 6(y + h) , 3x 3y y < 0. (3.5) where m:; = J j cjy P . . . 1 P •  = earth r e s x s t i v x t y = — oj = angular frequency y = permeability 6 = Difac function E^,E 2 = e l e c t r i c f i e l d above and below ground. Imposing the r e s t r i c t i o n of continuity of e l e c t r i c f i e l d at y = 0 for E^ and as boundary conditions, we can obtain a general expression 52 y 7 T T Figure 3.2: Conductor configurations for earth return formula. 53 for the underground e l e c t r i c f i e l d . We can then obtain the expression for the mutual earth-return impedance of underground cables by d i v i d i n g E 2(x,h 2) by the current i (O.h^) t o 8 e t Z. . = i j 2IT 0 0 / 2" 2* 0 0 '/2 2~\ exp(-£/ a +m ) & j ax ^ + j exp;(-l//a +m ) da - 1 1 ' n~~2 a + Va +m 0/2.2 2/a +m exp( -£Va 2+m 2) 2v42+m2 da (3.6) exp,(= v^/a2-rm2: lax 2^.00 1 a I +/a2+ra2 da + (K n(mDj - (K 0(mD 0)) 2 77 s ( T 1 0: 2' = A Z 4 J + 4 ^ fe-JuDj - K„(mD„)) J i j 1 2 7 7 ^ o ^ l CT 2y (3.7) where = Kelvin function of order zero m D, = = y ^ ^ " j x = h o r i z o n t a l separation between cables /x /2 2 D 2 = /x + ( ^ 4 - ^ ) h^,h 2 = depths of b u r i a l of cables a = l = |h 1 + h I AZ.. = Carson's Correction term, i d e n t i c a l to that f o r over- head l i n e s case. The above formula i s also applicable to s e l f earth return impedance components. In such cases, the terms f o r and can be re-defined as D 1 = r D 2 = 2h (3.7a) • where r i s radius, h = depth of buried cable. However, the above formula i s unsuitable f o r st r a i g h t forward c a l c u l a t i o n s . Before d i g i t a l computers are widely used, approximate r e s u l t s were obtained 28 only a f t e r c e r t a i n l i m i t i n g conditions were accommodated. Then, Wedepohl et a l used Cauchy's integration for the Carson's correction terms and derived approximate formula f o r the equation given i n equation (3.7) as mD „ Z. . = ^ { £n(Y-±-) + i - f mA } (3.8) i j • 2 ir 2 2 i for |mD | < 0.25 and Y = Euler;'s constant = 0.5772157. However, the above formula gives r e s u l t s which are about 20% higher when compared with the d i r e c t numerical computation using the o r i g i n a l equation as i n Ametarii's case. The i n f i n i t e i n t e g r a l for the underground cable i s the same as the Carson's correction terms for overhead l i n e s . We can define a new parameter a as a = / ^ D = 4TT /5 10 4 D / | - (3.9) 55 where D and p are in MKS units 2h. for s e l f earth return impedance ^ ~ for mutual earth return impedance This correction term i n t e g r a l can be represented by the following . .„ . . 3 4 i n f i n i t e converging s e r i e s : 1. For a < 5 AR' = 4o)-10~4{^-o -b^a* cost)) 2 2 +b^[ (c2~lna)a cos2<{)+(j)a sin2cj)] 3 +b^a cos3<}> -d^a4cos4cf> -b^a^cos5<j> 6 6 +b^ [ (c^-lna)a cos6<j)+cba sin6c))] +b^a^cos7<j> g -d Qa cos8<f> o - ...} AX' = 4to-10 4{|<0.6159315-lna) +b^a*cos<j> 2 -d^a cos2cf> 3 +b^a cos3<j> 4 4 -b^[(c^-lna)a cos4cf>+(j>a sin4cf>] +b^a^cos5cj) -d^a^cos6tj) +b^a cos7<j> 8 8 -b Q[ ( c Q - l n a ) a cos8<J>+c(>a sin8c()] o o + ...} 56 Notice that each 4 successive terms form a r e p e t i t i v e pattern. The co- e f f i c i e n t s b., c. and d. are obtained from the recursive formulas: 1 1 l /2 ^ b^ = — - r f o r odd subs c r i p t s , b. = b. „ . ẑfoN with the s t a r t i n g value -\ l i-2 i(i+2) 1 ^•b„ = f ° r even subscripts, 2 I D c = c. „ + + -rrr with the s t a r t i n g value c„ = 1.3659315, l i-2 l i+2 - 2 A 7 1 U d. = j- • b., l 4 l with sign = ±1 changing a f t e r each 4 successive terms (sign = ±1 for i 1,2,3,4; sign = -1 for i = 5,6,7,8 e t c . ) . 2. For a > 5 , _ ' cos((> /2 cos2<)) cos3<}> 3cos5<j) 45cos7(j) - ( 2 3 5 " 7 ) 4a)-10 4 /I Q3.ll) -4 AX1 - ( C O S 1 ^ _ cos3<l> + 3cos5<{> + 45cos7(}> ^ # 4ai* 10 3 5 7 >=-a a a a v2 It should be noted that the correction terms w i l l become zero when the parameter a i s very b i g , i e . when frequency or cable distance from g.round i s very large or when earth r e s i s t i v i t y i s very small. The Kelvin functions can also be calculated by another i n f i n i t e 36 converging series as can be obtained from availabe source. , ''-It can also 35 be calculated by a s p e c i a l subroutine CBESK ava i l a b l e from the UBC Computing Centre. A f t e r e s t a b l i s h i n g the numerical formula for cable earth return impedance, the discrepancies between the r e s u l t s of Wedepohl et a l and those of Ametani can then be c l a r i f i e d . The author has confirmed that accurate cable earth return impedance •can be calculated by using d i r e c t computation of i n f i n i t e series sub- 40 s t i t u t i o n f o r i n f i n i t e i n t e g r a l and Kelvin function. Cable mutual impedance from Ametanis computation i s acceptable though d i f f e r e n t i n f i n i t e s e r i e s i s used for the Carson's correction terms. I d e n t i c a l r e s u l t s are obtained at l e a s t to 4 s i g n i f i c a n t figures for frequencies up to 100 k Hz (See Figure 3.2). The approximate formula given by equation (3.8) on the other hand, gives r e s u l t s about 20% c o n s i s t e n t l y higher. The author has also confirmed that the earth return impedance for underground cable can be approximated by the equivalent earth return impedance for overhead l i n e s . The expression for earth return impedance for overhead l i n e i s D Z.. = to -==- + AZ.. , .„ i j 2TT DJ^ I J ' (3.12) for mutual earth return impedance 2h and Z. . = to — + AZ. . , (3.13) xj 2 It GMR xj for s e l f earth return impedance where i s distance between i * " * 1 and image of j 1 " ^ conductor; i s distance between i * " * 1 and j t ' 1 conductor; • h i s height of conductor above ground; GMR i s geometric mean radius= radius of conductor at high f r AZ i s Carson's correction term. I d e n t i c a l r e s u l t s up to 3 or 4 figures are obtained for earth r e s i s t i v i t y of 1 to 100 Q-m up to the frequency of 100 K Hz. The 58 consistency between these two r e s u l t s i s due to the fact that the Kelvin functions K (/j" x) = ker(x) + j kei(x).; 36 can be evaluated by the following convev^ing series for 0 < x < 8: ker x = — In ( i r ) berz+Jxbei x—.57721 566 1 -59.05819 744(jf/8)*+171.36272 133(x/8)8 : -60.60977 451(z/8)12+5.65539 121 (x/8)'8 -.19636 347(x/8)20+.00309 699(:r/8)24 -.00002 458(2-/8)28+« (3.14) H<ixio- 8 kei*=-ln($x)beia;-fir ber a,+6.76454 936(z/8)* -142.91827 687(i/8)6+124.23569 Q50(z/8y° -21.30060 904(ar/8)14+l. 17509 064(r/8)18 -.02695 875(ar/8)22+.00029 532(x/8)29+« _(3.15) |€|<3X10-» where ber x=1 -64(as/8)4+113.77777 774(*/8)8 -32.36345 652(z/8)12+2.64191 397(a;/8)16 -.08349 609(*/8)M+.00122 552(x/8)24 -.00000 90l(x/8)28+e |«|<ixio-» (3.16) bei x=16(a;/8)2-113.77777 774(x/8)* + 72.81777 742(a/8)l0~ 10.56765 779(x/8)u + .52185 615(z/8)l8-.01l03 667(ay8)22 +.00011 346(z/8)29+e M<8X10-» Figure 3.2: Mutual impedance between outermost cables by d i f f e r e n t computation methods. 60 and x = ^f- D, (3.18) u = 4TT x 10 ^ ^m oi = frequency D = distance Di o r D L 2 P = earth r e s i s t i v i t y For frequencies up to about 1 M Hz and earth r e s i s t i v i t y of about 100 ft-m,and cable separation or cable depth of about 1 m, the term x i s r e l a t i v e l y small as x -> 0 Then, one can rewrite equations (3.14) to (3.17) for x 0 as ber = 1 b e i = 0 • • K Q(/fx) = ker x + j k e i x = - £n |- x - 0.57721 - j J (3.19) Thus, for the earth return impedance as shown i n equation (7), we have Z. . = AZ. . + ™ (K (mD.) - K (mD.)) xj xj 2 IT o 1 o 2 mDn - 4 Z« + if « - «• -r - - jj) - mD ( - in -f - - .57721 - jJ-)) = AZ . + ^ • in ̂  , x * 0 y 1 J 2 l T j , °1 Carson's correction term self-term which i s the same as i n equations (3.12) and (3.13). 61 The numerical r e s u l t s f or the self-term component of the s e l f and mutual earth return impedance for the underground cable obtained by the d i f f e r e n t formulae developed e a r l i e r are shown in Table 3.1. As can be seen from Table 3.1, the r e s u l t s obtained by these d i f f e r e n t formulae are very consistent. The f i n a l r e s u l t s f or mutual impedance from these methods are also shown i n Figure 3.3 for frequencies up to 1 M Hz. Because of the observed consistencies, the overhead l i n e formula approximation i s therefore recommended f o r underground cable for a l l frequencies up to 1 M Hz and earth r e s i s t i v i t i e s above 1 ft-m. 4. Calculation of s e l f impedance matrix for single core cable After the s e l f and mutual impedance for earth return of under^. ground cables,.is obtained, the s e l f impedance of i n d i v i d u a l cables can be calculated and the obtained r e s u l t s f or d i f f e r e n t current loops can then be transformed to the required form for the impedance diagonal submatrix Z as shown i n equation (3.1). s At f i r s t , one can consider the current in each of the i n d i v i d u a l cables flow i n two adjacent loops as shown in Figure 3.4. Loop 1 i s formed by the current flowing through the core and returning through the outside sheath. Loop 2 i s formed by the current flowing through the sheath and returning through the outside earth. These two loops can be described by equation (3.21) as d v l dx Z l l Z12 4 1 1 d v 2 dx _ Z21 Z22 _ i 2 _ where Z = Z by symmetry. 62 Separation i n meter D l D2 (1,3) .889 2.189 .901+j.OOO (60Hz) .901 .895-J.020 (100Hz.) .901 (1,2) 1.778 2.676 .408+j.OOO (60Hz) .408 .403-J.018 (100Hz) .408 (s e l f ) .254 2.0 2.064+j.OOO (60Hz) 2.064 2.057-J.022 (100Hz) 2.064 where f = 60 Hz, m = / j 6 ° X 2 \ l Q ^ X 1 0" 7-= .0022 /J i TT /icoy /jlOO x 2TT x 4TT X 10 ^ A Q n rr f = 100 Hz, m = / ^ = / J JOO -089 / j underground cable (exact) Z i j = T r f ( V m V ' K 0 ( m D 2 » + A Z i j overhead cable (approximation to above) hi " ^ *> 57 + * Z 13 • - - » * 0 Table 3.1: Mutual and s e l f earth return impedance terms as given by under- ground and overhead cable formula. underground overhead Figure 3.3: Approximations of mutual impedances between underground cables by Carson's formulae. Figure 3.4: Current loops inside SF^ cable f o r s e l f impedance c a l c u l a t i o n . 65 The matrix elements of equation (3.21) can be obtained by considering the i n d i v i d u a l current loop components making up the corresponding loops 1 and 2 as Z = Z + Z + Z 11 core-outside core/sheath i n s u l a t i o n sheath-inside (3.22) Z = Z + Z 22 sheath-outside earth-inside (3.23) z = z = —z 12 21 sheath-mutual (minus sign since i and ± i n d i f f e r e n t d i r e c t i o n ) . (N.B. Z ^ i s n e g l i g i b l e when sheath thickness » skin depth) where the i n d i v i d u a l elements are (3.24) (Zl) Z core-outside (Z2) Z core/sheath i n s u l a t i o n (Z3) Z sheath-inside (Z4) Z sheath-outside (Z5) Z earth-inside i n t e r n a l impedance of core with return through outside (sheath). impedance of SF, i n s u l a t i o n due to the o time varying magnetic f i e l d . i n t e r n a l impedance of sheath with return through ins i d e (core). i n t e r n a l impedance of sheath with return through outside (earth). s e l f earth return impedance, t h i s can be calculated by equations (3.7) & (3.7a), or can also be obtained by equation (3.25) with the approximation of i n f i n i t e 33 outside radius (Z6) Z sheath-mutual = mutual impedance of tubular sheath between loop 1 i n inner surface and loop 2 i n outer surface of sheath. 66 The i n d i v i d u a l s e l f and mutual impedance terms can be again obtained by solving the Maxwell's equations for the coaxial conductors as s i m i l a r to equations "£3.2)&(3 .3) # They are a function of frequency as derived by 37 28 Schkelkunoff and Sunde as . . , = (I n(mq) K, (mr) + K„(mq) I.,(mr)) tube-inside 2iTqp 0 1 0 1 (3.25) tube-outside = 2 ~ - (I Q(mr) KL(mq) + K Q(mr) I^mq)) (3.26) Jtube-mutual with 2Trqrp p = I 1(mr) ^(mq) - I (mq) K (mr) (3.27) (3.28) where and y q r m = angular frequency = 2i:f permeability = P ^ Q J y r = 1 for A l outside radius of tubular conductor inside radius of tubular conductor p = d.c. r e s i s t i v i t y V r i K0''K1 p Bessel functions Kelvin functions A f t e r obtaining the i n d i v i d u a l terms of the loop equation matrix as shown in equation (3.21), we can then obtain the diagonal sub-matrix elements by applying the following c i r c u i t conditions: V = V - v 1 c s V = V 2 s (3.30) 1 = 1 1 c (3.31) i» = i + i 2 c s (3.32) The i n d i v i d u a l loop equations of equation (3 21) then becomes dV dV -T9" + -J^ = ( Z n + z i , ) i + Z i o 1 dx dx 11 12 c 12 s dV and — = (Z + Z,_) i + Z„„ i dx 12 22 c 22 s (3.33) (3.34) Adding equation (3.33) to (3.34) gives dV dx ( Z11 + 2 Z12 + Z22> \ + ( Z12 + Z22) is ' ( 3 " 3 5 ) Thus, we can rewrite the s e l f sub-matrix Z as ' s Z 1 1 + 2 Z 1 2 + Z 2 2 Z 1 2 + Z 2 2 dV c dx dV" s dx Z12 + Z22 J22 (3.36) 5. /ffheath current refragn c h a r a c t e r i s t i c s for, usual earth As current flows along the core of the buried SF^ bus, a return path i s formed on i t s own sheath and possibly also on the surrounding s o i l and adjacent sheaths. Whether a l l the currents w i l l return through i t s own sheath depends s o l e l y on the frequencies involved. Due to the skin e f f e c t i n sheath material (Aluminum), a l l core current w i l l return through i t s own sheath for frequencies above 1 k Hz. In r e a l i t y , the SF^ cable i s l a i d on the ground surface (e.g. inside the lead shaft) or i s constructed above ground and grounded at c e r t a i n i n t e r v a l s (e.g. inside the substation). This cable l o c a t i o n even favor more current returning through the sheath than the ground as compared to buried'cable case. One can thus in v e s t i g a t e the l i m i t i n g case with the cable buried underground. This cable l o c a t i o n w i l l favour l e a s t 68 core current returning through i t s own sheath. In order to investigate the sheath current return c h a r a c t e r i s t i c s and therefore the mutual coupling between cables, one has to use the seri e s impedance matrix. One can consider cases i n which adjacent sheaths are e i t h e r included or excluded. a. Sheath current return c h a r a c t e r i s t i c s for single cable system For t h i s case, one only has to consider the s e l f diagonal submatrix of the series impedance matrix as dV dx dV dx J s l l J s l 2 J s l 2 Js22 (3.37) Since the sheaths of the three i n d i v i d u a l SF^ cables are s o l i d l y grounded at short j o i n t i n t e r v a l s or l a i d on earth surface, or buried inside .the earth, the sheath voltages can be considered to be zero for a l l p r a c t i c a l purposes. Then Equation (3.37) becomes 0 J s l 2 Z or s!2 's22 i + Z c s22 I -c At high frequencies as sheath mutual impedance i s n e g l i g i b l e ^ 3 3 ) when sheath thickness greater \ t h a n skin depth( See Appendix A.). Thus, neglecting the other two sheaths, the current return c h a r a c t e r i s t i c s of the SF,, cable through i t s own sheath from the core o can be calculated as i n Equation (3.38). The obtained r e s u l t s are shown i n Figure 3.4 .. For t h i s case, e s s e n t i a l l y a l l the current through the core w i l l return through i t s own sheath above the frequency of 10 Hz. 69 "?"sheath core z 0 sheath-mutual '''sheath """core earth _ single-cable system three-cable system: -current i n one core • ' -current i n three cores ® - i B l / i c l ® " i s 2 / l c 2 ® " 1 s 3 / l c 3 p = lOOft-m depth = 0 or .254m log frequency 1 — 10 100 l k 10k (Hz) Figure 3.4: Ratio of core current return through own sheath for single-and three-cable system. 70 In other words, the sheath acts as a perfect magnetic s h i e l d above the frequency of 10 Hz. Because of t h i s consideration, a l l SF^ cables are decoupled from one another and can be represented as 3 s i n g l e phase systems. I t i s also found that a change i n depth (1 m to .254 m) of cable does not change the current return c h a r a c t e r i s t i c s noticeably. b. Sheath return c h a r a c t e r i s t i c s f o r 3-cable system Since a l l the 3 sheaths of the SF^ bus are s o l i d l y grounded, the current w i l l return through a l l the three sheaths at lower frequencies (< 60 Hz). At higher frequencies, however, a l l the core current w i l l return through i t s own sheath because of the skin e f f e c t on the sheath. Here, again, one can conclude that the three SF^ cables are decoupled from one another. For t h i s case of 3-cable system, one can also consider the sheath voltages to be zero. One can substitute t h i s condition into equation (3.1) and obtain dV " c l dx - 5 s i i Z s l 2 Zml2 Zml2 Zml3 Zml3 ^ 1 0 Z s l 2 Zs22 Zml2 Zml2 Zml3 Zml3 ^ 1 d V c 2 dx Zml2 Zml2 Z s l l Z s l 2 Zml2 Zml2 \2 0 Zml2 Zml2 Z s l 2 Zs22 Zml2 Zml2 is2 d V c 3 dx Z'ml3 Zml3 Zml2 Zml2 Z n s l l Z s l 2 c3 0 Zml3 Zml3 Zml2 Zml2 Z s l 2 Zs22 s3 (3.39) (N.B. Z',„ = Z as symmetrical arrangement of cables as ml2 mz 3 i n Figure ,3.1b) Equating the zero sheath voltages f or the 3 cables, we have 0 = Z c 1 0 i . + Z „ i , + Z. . . ( i + i ) + Z (1 + i ) (3.40) sl2 c l s22 s i ml 2 c2 s2 ml 3 c3 s3 0 = Z 1 0 ( i . - . + i ) + Z i . + Z i + Z 1 0 ( 1 . + i ) (3.41) ml 2 c l s i s l 2 c2 s22 s2 ml2 c 3 s3 0 = Z , 0 ( i - + i n) + Z. , 0 ( i „ + i 0) + Z _ i _ + Z 0 „ i _ ,~ / 0 . ml3 c l s i ml 2 c2 s2 s l 2 c 3 s22 s3 (3.42) If we assume phase B i s energized, i . e . , we assume ± c l = i c 3 (3.43) ^ 1 = S 3 ( 3 - 4 4 ) i t l = i c 3 = 0 (3.45) Then, substitute equations (3.43) to (3.45) into (3.40), we obtain 0 = Z c 0 0 i + Z i + Z i + Z i S 2 2 s i ml2 s2 ml3 s i m!2 c2 = (Z. ,n_ + Z ) i + Z. i + Z i - ml 3 s22 s i ml2 c2 ml2 s 2 Znil3 + Zs22 X s l , 1s2 , / 0 or + = -1 (3.46 ml2 1 c 2 1 c 2 Also substitute equations (3.43) tO (3.45) into (3.41), we obtain 0 " 2 Zml2 \l + Zs22 \2 + Z s l 2 ! ! i ^ . i 5 l + ! 5 2 2 ^ = _ ± (3.47) Z s l 2 Xc2 Z s l 2 1 c 2 72 By solving equations (3.46) and (3.47), we get Zml3 + Zs22 ^ml2 2Z ml2 J s l 2 Zml3 + Zs22 Jml2 2Z ai2 'sl2 -1 -1 (Z ... + Z 0 0 ) Z - 2Z ml3 s22 sl2 ml2 Jml2 Js22 Z s l 2 2 Zml2' Zmi2 Z B 2 2 ( Zml3 + " Z s22 ) (3.48) -1 -1 Zml3 + Zs22 ml2 2Z nil2 J s l 2 Js22 Jsl2 -Z sl2 Js22 J s l 2 2Z m!2 Zml2 + Zs22 Jml2 Zml2 Zs22 ( Zml3 + Z s 2 2 ) (3.49) The r a t i o of currents i n sheath to core of phase B i s calculated and plotted as a function of frequency i n Figure3.4. The r e s u l t shows that at frequencies above 1 kHz as i n l i g h t n i n g surges, a l l current through the core w i l l return through i t s own sheath. Thus, one can conclude that the earth return current component i s not important and therefore the mutual 73 c o u p l i n g between cables can be i g n o r e d . c. Sheath current return c h a r a c t e r i s t i c s f o r 3 cable system with current i n a l l 3 cores For a case when currents flox^ i n a l l the three cores of the three SFg cables, the return current through sheath w i l l change accordingly. This suggests that such s i t u a t i o n s must also be investigated to deduce the mutual coupling e f f e c t among cables. Using the same equations as derived i n Equations (3.40) to (3.42), one can now put in currents i n the 3 cores by assuming and i c 2 - i . o IR i c 3 = 1.0 /120° i = 1.0 /-120' c l ' (3.50) (3.51) • (3.52) Rewriting equations (3.40) to (3.42) as Z S 2 2 \l + Zml2 ^ 2 + Zml3 ^ 3 = ' Z s l 2 \l " Zml2 ^ 2 " Zml3 ^ 3 = h <3-53) Zml2 \l + Zs22 S 2 + V ? ^ 3 " " Z,12 S i " Z s l 2 ±c2 " Zml2 S 3 = A2 ( 3 " 5 4 ) Zml3 S i + Zml2 S 2 + Zs22 S 3 = " Zml3 S i " Zml2 S 2 ~ Z s l 2 S 3 = A3 ( 3 ' 5 5 ) Defining the determinant T as Zs22 Zml2 Zml3 Zml2 Zs22 Zml2 Zml3 Zml2 Zs22 Zs232 + 2 Zml2 ' Zml3 " Zs22 ( Zml3 + 2 Z m 1 2 > (3.56) 74 We can then obtain the current through the three i n d i v i d u a l sheaths as' 2 2 2 i s l = A l Z s 2 2 t A3 Zml2 + A 2 Z m 1 2 Z m l 3 " A3 Zs22 Zml3 ~ A2 Zml2 Z s22 " A l Z m l 2 T (3.57) 2 2 i 2 = A2 Zs22 + A l Z m l 2 Zml3 + A3 Zml2 Zml3 ~ A2 Zml3 ~ A3 Zml2 Zs22 " A l Z m l 2 Z s 2 2 T - (3.58) 2 2 2 1S3 " A3 Zs22 + A2 Zrql2 Zral3 + A l Z r a l 2 " A l Z m l 3 Zs22 " A2 Zml2 Zs22 " A3 Zml2 T (3.59) After s u b s t i t u t i n g the conditions, for the 3 phase currents from equations (3.50) to (3.52) into equations (3.57) to (3.59), one can obtain the return currents through a l l i n d i v i d u a l sheaths. The magnitudes of the sheath currents are also shown i n Figure 3.4.it i s again confirmed here that at frequency above 60 Hz, a l l current flowing from core w i l l return through i t s own sheath. Each core i s completely shielded from the adjacent cores. Thus, the three SF^ buses are completely decoupled from one another and should therefore be represented by si n g l e phases as i n the case of l i g h t n i n g overvoltage propagation. 6. Sheath current return c h a r a c t e r i s t i c s f or substation earth with grounding g r i d network In r e a l i t y i n the substation, the cable sheaths are grounded inside the substation with a grounding network g r i d c o n s i s t i n g of copper bars which are connected across the whole substation. These grounding copper bars serve to reduce s i g n i f i c a n t l y the i n s i d e earth r e s i s t i v i t y of the substation. This suggests that the sheath current return c h a r a c t e r i s t i c s of t h i s reduced earth r e s i s t i v i t y should also be investigated as a reduction 75 i n earth r e s i s t i v i t y w i l l favor more current returning through the earth. The r e s u l t f or the sheath return current as a function of earth r e s i s t i v i t y i s shown i n Figure 3.5. This f i g u r e shows that the sheath return current increases as the earth r e s i s t i v i t y increases. This agrees with the manufacturers and u t i l i t y companies of SFg substations who claim that current returning through sheath inside the substation i s l e s s than 75%. Based upon t h i s c r i t e r i a , a nominal earth r e s i s t i v i t y of 0.3 x 10 ^ftm i s chosen. A f t e r choosing a nominal value for earth r e s i s t i v i t y , the sheath current return c h a r a c t e r i s t i c i s then evaluated as a function of frequencies and depth, as shown in Figure-3.6. Fluctuations in o v e r a l l sheath current r e s u l t s are shown i n Figure 3.6. At about 1 K Hz, the sheath current i s even found to be larger than the core current. This can be explained by the phasor diagram as shown i n Figure 3.7. In Figure 3.7,only m u l t i - cable systems with current i n centre core are shown, but mutli-cable system with currents i n a l l 3 cores would Kralso give i d e n t i c a l r e s u l t s . The present study again confirmed that a l l cores are decoupled from one another above 2 k Hz even for the adverse case of s i g n i f i c a n t l y reduced earth r e s i s t i v i t y inside the substation. It should be noted that the mutual impedance between cores, between sheaths and between corresponding cores and sheaths are a l l assumed to be equal by Wedepohl and Ametani. The s h i e l d i n g e f f e c t of the sheath i s neglected. The v a l i d i t y of t h i s assumption i n cable parameter computations could be the topi c of further research. I t i s of l i t t l e concern for the purpose of t h i s t h e s i s . At the high frequencies encountered i n l i g h t n i n g surge studies, core current always return completely through the sheath. In that case, the magnetic f i e l d becomes zero outside the sheath anyhow. log earth resistivity.*; 6 7 8 9 10 Figure 3.5: Ratio of core,current return through sheath at 60 Hz for d i f f e r e n t earth r e s i s t i v i t i e s . 20 30 40 (10"6ft-m) 77 sheath 120 100 80 60 40 z o sheath-mutual sheath earth core single-cable system three-cable system i „/i s2 c2 (§) : depth or height = .254 m (E) : depth or height = 1 m © : depth or height = 6 m p = 3uft-m 20 log frequency i i 1 1 • 10 100 l k 10k 100k (Hz) Figure 3.6: Ratio of core current return through own sheath for s i n g l e - and three- cable system at reduced earth r e s i s t i v i t y of 3 yft-m i n s i d e substation. 78 .1 Hz ->- i core """so i i more core current return through s o i l than sheath 60 Hz i core :". sheath more core current return through sheath than s o i l l k Hz 10k Hz -> ->- i core i core , - — — — ' 1 s h e a t h -y •+ I i J> i 1 sheath 1 c o r e 1 """sheath " c c o r e •"•core """sheath + """soil Figure 3.7: Phasor diagram of current return through sheath and earth. 7 • Formation of shunt admittance matrix for SFfi cables 79 For a usual 3-phase sing l e core underground cable system, one can b u i l d a 6 x 6 shunt admittance matrix [Y ] to describe the cable system as i . e . Ldx J [Y] [V] rdid~ y l - y l 0 0 0 0 dx s i " y l y l + y 2 0 0 0 0 • dx d i c 2 0 0 y l 0 0 dx I d l s 2 0 0 - y l y l + y 2 0 0 dx d i _ c3 0 0 0 0 y l - y l dx d i . s3 0 0 0 0 - y l y l + y 2 dx c l s i c2 V s2 c3 s3 (3.60) Notice that the off-diagonal submatrix of [Y] are a l l zero due to the fact that the grounded sheaths in between acts as e l e c t r o s t a t i c s h i e l d between cables. For the SF^ cable system as shown i n Figure 3.1, the diagonal submatrix elements are i y.. = iwc, = io) 2T T £ '1 J 1 J o Jin r3 y2 = ^ w c2 = 2 7 r e 0 * Jin where y^ i s admittance due to i n t e r n a l SF^ gas i n s u l a t i o n , and i s admittance due to external sheath i n s u l a t i o n . (N.B. ',Th e external i n s u l a t i o n i s non-existent for the SFg cable.) 80 As has been confirmed by the author i n previous f i n d i n g f or sheath current return c h a r a c t e r i s t i c s , the core should be represented as si n g l e phase. Then, the admittance equation shown i n Equation (3.60) should be reduced to d i C y V = J'l C dx In — r2 y s e l f * V c (3.62) 8. Confirmation of numerical accuracy for cable parameter c a l c u l a t i o n and current return r a t i o s The numerical accuracy of the computation was confirmed when the cable parameters obtained by the developed cable constants program, 32 and by the BPA cable constant program agree co n s i s t e n t l y to more than three s i g n i f i c a n t f i g u r e s . 38 Then, a 500 kV submarine cable was chosen as another t e s t example. In t h i s case, the cable parameters for the submarine cable was f i r s t c a l - culated. The amount of core current returning through the sheath, armour and the sea was obtained by taking into account of zero p o t e n t i a l s on the grounded sheath and the grounded armour. The r a t i o s of magnitudes of core current returning through the sheath ,the armour, and the sea at 60 Hz were obtained as 14%, 87.8% and 5.6% r e s p e c t i v e l y . These agreed to more 48 than two figures to the r e s u l t s of other fi n d i n g s . 9. Single phase representation parameters for multi-phase SF6 cables Since a l l phases of the SF^ cables are decoupled from one another, single phase c a b l e representation for studying l i g h t n i n g overvoltage wave 81 propagation i n SF, cable i s recommended. The s e l f admittance element (y ) " s e l f can be calculated from the simple formula as shown i n equations (3.61) and (3.62), whereas the s e l f impedance matrix element can be calculated from equation (3.1) as d V c l Z i + Z i + Z 1 0 ( i „ + i „) + Z . , ( i . + i J (3.63) — == s l l c l sl2 s i ml2 c2 s2 ml3 c3 s3 dx and 0 = Z s 2 1 i c l + Z s 2 2 i s l + Z m l 2 ( i c 2 + i ^ ) + Z m l 3 ( i c 3 + { y ^ Subtracting Equation (3.64) from (3.63), we get ( Z s l l " Zs21> S i + ( Z s l 2 * Z s 2 2 ) S i dx = ( z s i i " W 1 - TT> " W r r " S i > < 3 - 6 5 > c l c l = s e l f c l where the sheath to core current r a t i o can be obtained from equation (3.49) or equations (3.57) to (3.59). Because a l l core current returns through the sheath at high frequencies, one has i n such condition S i = -1, and Z = Z i , ' sl2 s22 . . c l Substituting t h i s into equation (3.65) or (3.37), one obtains dV dx ( Z s l l " Zs22> S ( 3 ' 6 6 ) After having obtained the s e l f s e r i e s impedance for s i n g l e phase cable as shown i n Figure 3...1, one can then represent the cable by the surge surge impedance Z , and the wave propagation v e l o c i t y v, as given by 82 zsurge / ! s e l f = ^ fl ( 3 > 6 ? ) / y s e l f v = oi / - — = 300 m/ys (3.68) / s e l f • y s e l f One should r e a l i z e that the m e t a l l i c sheath of the SF^ cable always form a very good earth return path to the cable core. The seri e s r e s i s t a n c e i s n e g l i g i b l e compared tp the reactance (See Fig.3,8),Thus, the SF^ cable can be taken to be l o s s l e s s . I t should also be noted that f or such a simple go-return c i r c u i t f o r a co a x i a l cable, the inductance can be given by the simple f o r m u l a ^ as y r L = •—- in — (3.69) - • 2TT r 2 = 0.205 VH/m Consistent r e s u l t s f o r the inductance are obtained by equation (3.65) and (3.69) for frequencies above 10 Hz. Thus, the simple formula i s i n equation (3.69) i s recommended for inductance c a l c u l a t i o n of SF^ cable i n the study of surge propagation c h a r a c t e r i s t i c s . The surge impedance and 7 the wave propagation v e l o c i t y can be then obtained as surge = /1 = A ° .. £ n I 3 . _1_ . £ n I 3 C y - 2TT r 2 2 T T E o r 2 ° 1 „ 3 e , 2 r„ o 4TT 2 r 3 = 60 £n — (3.70) r2 where r^ and r 2 are radius of sheath and core respectively. 10 Figure 3.8: Self series inductance, resistance and resistance to reactance r a t i o for SF, cable. 84 and v = /f^ = / — J i n — (3.70) •' T r / y r„ 2T T E r„ r2 y £ o o = 300 m / y s v e l o c i t y of l i g h t i n vacuum 10. Wave propagation i n SF6 cables Wave propagation c h a r a c t e r i s t i c s i n sing l e core SF^ cable can now be modelled by the surge impedance of 61.4 ft,(typically about 60 to 75 ft), and wave propagation v e l o c i t y , ( t y p i c a l l y 300 m / y s ) . A numerical simula- ti o n of overvoltage wave-shape .in the receiving end of a SF^ cable j o i n i n g to a overhead transmission l i n e i s simulated. The r e s u l t i n g voltage i n the open-circuited SF^ cable receiving end r i s e s i n a s t a i r c a s e fashion of diminishing amplitude, to a value of 2 p.u. (See Figure 3.9). This can be explained by using the r e f l e c t i o n (C.) and re f r a c t i o n c o e f f i c i e n t (C ) of the system at the li n e - c a b l e junction and is. 39 the open-circuited cable end re s p e c t i v e l y . For the l i n e - c a b l e junction at A, one has Z 2 = 312, Z1 = 60ft Z - Z (Wave incident from cable) c ^ = z + 7^ = 312 '+" 60 = ( 3 - 7 ± ) (Wave incident from l i n e ) Z 2 = 60, Z = 312 ft 85 Figure'3.9: Overvoltage waveshapes at both ends of SF^ cable j o i n i n g from overhead transmission l i n e . 86 For the open end of the cable, we have 2 1 °° - 60 °° + 60 = 1 and C ,o R 2 x °° + 60 2 (Wave incident from cable, Z = 60, Z = ») Thus, the discr e t e r i s e i n voltage wave shape can be expressed as v = 2 x .32 (1 + C.+ C. 2 + . . .) where each step a d d i t i o n accurs at di s c r e t e time i n t e r v a l s of 2 t r a v e l times. On the other hand, t h i s o v e r a l l r i s e i n overvoltages wave shape also agrees with the general exponential r i s e wave shape i n charging of a capacitor. This i s due to the inherent large s e l f capacitance of cables. The o v e r a l l r i s e i n wave shape can : be sketched by modelling the SFg cable as a lumped capacitor equivalent to the t o t a l capacitance for the length of cable, and ignoring the surge impedance of the cable (See Figure 3.9). 87 CHAPTER 4: CORONA ATTENUATION AND DISTORTION CHARACTERISTICS OF LIGHTNING OVERVOLTAGE IN OVERHEAD TRANSMISSION LINES. 1. Introduction As the l i g h t n i n g voltage wave t r a v e l s down the overhead transmission l i n e , a high e l e c t r i c f i e l d i s produced on the l i n e conductor surface. When kV the e l e c t r i c f i e l d i n t e n s i t y exceeds the breakdown strength of a i r (^30 /em), i o n i z a t i o n of surrounding a i r molecules takes place. This phenomenon w i l l d i s s i p a t e the unwanted surge energy away from the system and thus reduces the magnitude and i n i t i a l rate of r i s e o? t h e l i g h t n i n g overvoltage. In transient l i g h t n i n g overvoltage studies, several numerical methods 44 have been employed to account for corona e f f e c t s . Brown applied the concept of corona radius to account for the corona envelope produced on the conductor surface. The coronated l i n e capacitances at higher voltages are also obtained '• o 12 by extrapolation. Darveniza also used lower wave propagation v e l o c i t i e s higher voltages and d i f f e r e n t corona correction factor for d i f f e r e n t conduc- tor configuration. However, both methods are not straightforward and are 43 not t o t a l l y successful i n d u p l i c a t i n g f i e l d rest r e s u l t s . Umoto and Hara also transformed the transmission l i n e equation for coronated l i n e s into difference algebraic equations. However, t h i s numerical approach i s not e f f i c i e n t enough. Thus, an e f f i c i e n t and accurate numerical model for corona must be developed to predict the corona attenuation and d i s t o r t i o n character- i s t i c s on l i g h t n i n g overvoltage propagations i n overhead l i n e s . 2. Physical properties of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s The p h y s i c a l aspects and laws governing the behaviour of corona d i s - charge have been investigated since the beginning of t h i s century. However, most of the investigations and a p p l i c a t i o n s have been l i m i t e d to power frequency steady state or at most to switching transient conditions. From 88 the published f i e l d measurements f o r l i g h t n i n g surges, i t can be observed that the attenuation r e s u l t i n g from corona e f f e c t s i s much larger than that r e s u l t i n g from transmission l i n e s e r i e s resistance losses. The non-linear c h a r a c t e r i s t i c s of the corona discharge can be considered as (see Figure 4.1): a) Corona attenuation loss - From the quadratic law of corona loss pro- 41 posed by Peek , the loss (v i^) per un i t length i s proportional to the square of the voltage above the c r i t i c a l corona voltage v i . e . where k = a • /-^r x 10 /m r , h = radius and height of conductor r e s p e c t i v e l y a = Corona loss constant determined experimentally This corona attenuation l o s s can be modelled with a r e s i s t i v e current l o s s i ^ through the corona r e s i s t i v e branch to ground as b) Increase i n shunt capacitance - the retardation of the wave front by 42 corona can be explained by an increase i n shunt capacitance. S k i l l i n g 43 and Umoto suggested that the increase i n shunt capacitance i s proportional to the voltage above the c r i t i c a l voltage V c q , i . e . V C = 2k (1 - (4.3) corona c V where k = a x 10 h c c v 2h a = corona los s constant^determined experimentally c 89 V - V R - v c transmission l i n e Corona shunt capacitance V / C = 2k (1 „ corona c V. 1 + K c R G =kn--~f corona R V rrfn rmr Figure 4.1: Nonlinear corona losses model. 90 This increase i n capacitance can be modelled by : a capacitance branch to ground with the capactive current los s i„ Corona discharge only occur i f the voltage i s greater or equal to the c r i t i c a l corona voltage, and i f the voltage increase with time, i . e . j 3v v £ v , and — > o. co 3t This i s due to the fact that, when the voltage begins to decrease, the space charge c o n s i s t i n g of heavy ions i n the i o n i z a t i o n region remains p r a c t i c a l l y constant i n magnitude and p o s i t i o n during a short period of time. This slow d i f f u s i o n of ions r e s u l t s i n l i t t l e energy loss i n the case of decreasing voltage conditions even when v > V C Q . 3. Transmission l i n e equations f o r coronated l i n e s . The corona phenomena can now be described by the modified l i n e equations. With the introduction of d i g i t a l computers, these phenomena can be studied accurately by solving the equations describing the electromagnetic wave propagations taking corona into account as follows: = £ + ( i - ^ S + v1-'^2-' <4-6> extra shunt Extra shunt capacitance conductance due to due to corona corona 43 45 Umoto and Inoue solved the above equations by the d i f f e r e n c e method. The l i n e equations (4.5) and (4.6) are transformed into algebraic equations of small increments of distance, Ax, and time, At. However, t h i s method i s not e f f i c i e n t to implement into the d i g i t a l computer as the method requires Ax to be as small as 7 m r when using At = .01 ys. 4. Solution of l i n e equation by compensation method with trapezoidal rules The l i n e equations (4.5) and (4.6) with corona losses can be solved by the compensation method. In t h i s method, the l i n e equations are f i r s t solved without the extra corona terms. The Bergeron's method using t r a v e l l i n g wave technique together with modal ahalysis(See Chapter 2) i s a p p l i e d . Then, the corona losses can be treated as non-linear shunt branches connected to ground,. .< The trapezoidal r u l e can then applied to obtain the t o t a l current los s of the corona phenomena. By applying the trapezoidal rule of l i n e a r i n t e r p o l a t i o n to the corona r e s i s t i v e branch to ground, we have j as shown i n Figure 4.2, ± - e - v t + A t = i W + ( v t " i V where d = + — (4.7) v — v t + At t as d = slope of graph at time t . Also, d can be obtained by considering the equation (4.2) 92 Current i ( v ) ( Voltage v Figure 4.2: Linear i n t e r p o l a t i o n for resistance corona branch. 8v/9t=v voltage v or current i Figure 4.3: Linear i n t e r p o l a t i o n f or capacitive corona branch. 93 ( v - v )' t CO t R v. or co k^ v t + k„ — - 2k„ v R v ~"R co d i d = dv v 2 K. kR v 2 (4.8) Thus, we eventually have 1 . , ,v - 1 i v V t . + At " d \ + At + ( fc d ° (4.9) \ ' i t + At + V o (4.10) where v = v - v i , ( known from past h i s t o r y at time t) o t d t and ^R = a" (known from past h i s t o r y at time t) S i m i l a r l y , since the corona capacitive branch current loss i s given by 2 k a c , , 9v x = (v - v ) — v co 3t or 9v v i 3t 2k (v - v ) c CO f ( v , i ) Applying l i n e a r i n t e r p o l a t i o n of the 2 variables ( i . e . from f i r s t term of Taylors' s e r i e s ) , we have as shown i n Figure 4.3 f ( v , i ) = f ( v , i ) t+At + 3v (v t+At \ , 3f ' Vt> + 31 ^t+At " ± t ) (4.11) 94 3f 3v i (v v ) - v 2k t c C O - X , V C O 2 k c <vt - v c o ) 2 (4.12) and 3f 3i 2kj;v - v ) t * co 2H\ - \o> (4.13) Thus, we obtain f ( v , i ) t+At 3v 3t t+At v — v t+At t " A t v. 2k(v - v ) v t+At / t ^ t co / Re-arranging equation (4.14) w i l l give the l i n e a r i z e d equation as Vt+At = R c S+At + V l (4.15) where R = 1 + v i * . . co t At 2k (v - v )2 C t C O * v At t (known from past 2k (v - v ) , . „ . c t co hxstory) and 1 + v K' A«-co t At 2k (v - v )2 c t co . . V X V At co t t ( V t + 2k (v - v )2> from c t co past hxstory) Combining equations (4.10) and (4.15) for the voltages and currents in both corona r e s i s t i v e and ca p a c i t i v e branches by taking into account v = v_. = v c R and i = i + i„ c R we can obtain ( ^ + ^ ) v - ( i c + i R ) + ( ^ + ^ ) of or v = R ' i + k' (4.17) R c *R where R' and k* = R c + R k *R V l + R c Vo R Having the corona loss branches represented by a l i n e a r model as described in equation (4.17), the compensation method can then be applied to solve the transmission l i n e equations including corona losses. In the compensation method, the transmission l i n e i s f i r s t reduced to a Thevinin equivalent (See Figure 4.4) and i s described by vv= V Q + A2±, (4.18) where i s a negative number. Then, t h i s equation i s solved simultaneously with the l i n e a r i z e d equation for corona l o s s , as in equation (4.17). Thus, the r e s u l t i n g corona voltage and discharge current can be obtained as A m (m i s ground) A : Thevinin equivalent network for transmission l i n e without corona losses B : Nonlinear corona losses model v = R' i + k' Voltage v. v corona «• v =v +A i • » v k m o 2 Current i km i corona Figure 4.4: Compensation method f o r non-linear corona model 1corona - R' - A. (4.19) R'v - A-k* A o 2 and v corona • R1 - A. (4.20) 5. Influence on corona by adjacent sub-conductors i n the same bundle Extra-high voltage phase conductors are designed to consist of several sub-conductors bundled together in order to reduce corona losses. The e l e c t r i c f i e l d on a sub-conductor surface i s affected appreciably by the adjacent sub-conductors i n the same bundles. The corona phenomenon i s consequently influenced. The e l e c t r i c f i e l d on the sub-conductor surface due to the sub- 46 conductor i t s e l f i s given by max 2 IT e r o , Qr-= charge/length cv 2 IT e r o (4.21) where c = e f f e c t i v e capacitance/length v = voltage of conductor r = radius of sub-conductor However, for a bundled conductor with 4 i n d i v i d u a l sub-conductors, the maximum e l e c t r i c f i e l d i s given by^(See Figure 4.5) max Q1 ,- 1 2TT e + s72 + 2 -• s i n 45 u) -21 •- -2-TT e r o (4.22) 98 \ a x - -2̂ tT ( 1 + vfs } C V „, e f f e c t i v e where Q = ^ Figure 4.5: C r i t i c a l voltage c a l c u l a t i o n by evaluation of maximum e l e c t r i c f i e l d on a 4-conductor bundle. A f t e r the maximum e l e c t r i c f i e l d on the conductor surface i s obtained, the c r i t i c a l voltage for corona discharge can be computed by 6 equating the maximum e l e c t r i c f i e l d to 30 kV/cm or 3 x 10 v/m, . the e l e c t r i c breakdown strength i n a i r . A t y p i c a l c r i t i c a l voltage f o r a single conductor has been found to be 277 kV, and that for a 4-conductor bundle to be 558 kV. 6. Influence on corona by adjacent phase conductors Since the conductors i n each phase are mutually coupled to one another, voltages are always induced i n the adjacent conductors. Thus the maximum e l e c t r i c f i e l d on the conductor surface i s affected. However, due to the design of transmission l i n e s for extra high voltage l e v e l s , separating between phase conductors are u s u a l l y large compared with radius of i n d i v i d u a l conductors. This e f f e c t u sually change the o v e r a l l c r i t i c a l overvoltages by l e s s than 10%. But t h i s change i n c r i t i c a l voltage produces n e g l i g i b l e e f f e c t s on the o v e r a l l corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s on overvoltage wave (See Figure 4.6). 7. Optimal lumping locations and number of corona branch legs The equations with corona phenomenon i s now solved by the dbmbensation method with the corona l o s s legs lumped at a few places along the transmission•-• l i n e . However, the optimal locations and optimal number of lumped elements has to be determined. At f i r s t , 20 corona loss branches 70 m apart from one another were lumped between f i v e transmission towers. Then, a separation of 350 m between the corona loss branches was used. This increase i n separa- ti o n increased the deviation of the predicted wave shape from f i e l d measure- ments' appreciably (See Figure 4.7), from about 5 to 10%. This suggests that f i e l d measurements v ± t ± c a i = 303-kV (include e f f e c t of. c r i xca adjacent phase conductors)) v . . 277 kV (neglect e f f e c t of c r x t i c a l adjacent phase conductors) Figure 4.6: E f f e c t of adjacent phase conductors on corona losses. Figure 4.7: E f f e c t of lumping distances on corona. 102 the optimal separation should be about 70 m. The f i e l d measurement for a 4-conductor bundled was then simulated. In using the corona loss constants (for case of 1-conductor bundle) a = 30 c a- = 10 x 10 6 s l i g h t l y higher overvoltages were obtained. Then, a new set of corona constants (for case of 4-conductors bundle) a = 30 c a = 20 x 10 6 was used to give r e s u l t s consistent with those from f i e l d measurements (See Figure 4.8). F i n a l l y , the negative impulse overvoltage was also simulated f o r the 4-conductor bundle case. The corona loss i n t h i s case was found to be much less than the p o s i t i v e Impulse case. The corona los s constants were determined to be a = 15 c a = 10 x 10 6 With these sets of corona constants, the f i e l d t e s t measurement was again r e p l i c a t e d c l o s e l y (See Figure 4.9). 8. Overall numerical modelling f o r corona e f f e c t s The f i e l d t e s t r e s u l t s of corona attenuation and d i s t o r t i o n c h a r a c t e r i s t i c s on a 500 kV te s t l i n e were r e p l i c a t e d by the method Overvoltages(kV) Figure 4.8: P o s i t i v e impulse on 4- conductor bundle." Overvoltages(kV) 2000 FDUR-CONDUCTOR BUNDLE 1 2 3 4 5 6 f i e l d measurements a =15, a =10xl0 6 c G r a =15, a =5x10 c G Figure 4.9: Negative impulse on 4T conductor bundle. 105 developed e a r l i e r . This method examined corona c h a r a c t e r i s t i c s i n both single and bundled conductor l i n e s . From the performed study, one can conclude that the e f f e c t s of bundling of conductors i s e f f i c i e n t i n increas- ing c r i t i c a l corona voltage. Furthermore, influence of adjacent phase conductors i s n e g l i g i b l e on corona e f f e c t s . Thus, i t i s concluded that s i n g l e phase l i n e representation i s s u f f i c i e n t f o r corona studies. F i n a l l y , i t i s determined that separation between the corona los s leg can be lumped at 70 m without s a c r i f i c i n g a loss of accuracy on the predicted coronated waveform. A. reduction i n distance between corona legs w i l l not improve the accuracy of the simulated r e s u l t s . It should be noted that l i g h t n i n g strokes w i l l r a r e l y h i t more than one conductor at one time; thus corona phenomena have only been included for one conductor i n t h i s t h e s i s , rather than for a l l three phases simultaneously. CHAPTER 5: CONCLUSIONS 106 The attentuation and d i s t o r t i o n of l i g h t n i n g overvoltage waves on multi-phase transmission l i n e s and multi-phase single core SF^ cables i n compressed SF^ gas-insulated substations was studied. Corona e f f e c t s of l i g h t n i n g overvoltages on overhead l i n e s were also investigated. Available f i e l d t e s t r e s u l t s for corona e f f e c t s were duplicated to within 5% accuracy. Results obtained with the techniques developed by the author are 21 useful f or l i g h t n i n g i n s u l a t i o n co-ordination studies and other re l a t e d ,. 13,14,40 ^ , i t J studies . xhe l i g h t n i n g surge wave front can be calculated at any l o c a t i o n inside the substation, eg., inside the SF^ bus or at the trans- o former terminal. Based on the studies described i n the thesis the following recommendations are made for future i n s u l a t i o n co-ordination design studies: 1. Multi-phase untransposed l i n e s can be represented by single-phase l i n e models using s e l f parameters calculated at a high frequency of approximately 1 M Hz (See Table 2.4). Series resistance should be ignored. Frequency dependent e f f e c t s are not important f o r propa- gation over distances l e s s than 2 km. 2. Corona e f f e c t s are important i n reducing the magnitude and rate of r i s e of the incoming l i g h t n i n g overvoltage surge. E f f i c i e n t s o l u t i o n techniques using compensation methods are developed to solve the non- l i n e a r corona attenuation and d i s t o r t i o n phenomenon. 3. Multi-phase SF^ si n g l e core cables can be represented by s i n g l e phase cable models. Series resistance can be ignored. Cable parameter can be obtained with the simple formula for a go-return c i r c u i t f o r a coaxial cable with s u f f i c i e n t accuracy. 107 APPENDIX A: SKIN DEPTH ATTENUATION IN CONDUCTING MEDIUM WITH FINITE CONDUCTIVITY. This section shows that the core current return c h a r a c t e r i s t i c s through the sheath for the SF^ cable could be obtained by a d i f f e r e n t approach. From the Maxwell's equations i n a conducting medium, we h a v e ^ V x E = -jwu H (A.l) V x H = jtoeE + aE = aE, for good conductors (A.2) where a i s conductivity of medium. From equations (A.l) and (A.2), we can get 2 2 V x V x E = V ( V « E ) - V E = - V E (for homogeneous medium) = -joiyV x H = -jwyaE = -m2E where m = /jtoya = ^ ^ • /u>ya This equation i s i d e n t i c a l to the d i f f u s i o n equation with solutions — = E = E e - m Z , (A.3) a x o = E e-V"2~ Z • e - V ~ T Z (A.4) o = E e - j Z / 6 • e " j Z / 6 (A.5) o 108 where 6 = ~-. = - 1 7= skin depth Thus, the tangential e l e c t r i c f i e l d E or the tangential current density J x w i l l be attenuated by ̂  = -368 when the depth of penetration Z equals to the skin depth. For aluminum, we have the skin depth 6 as <5 = 7 = = (A.6) /irf (4ITX10-7) (3-8x10/) 8-1 ,— cm Therefore, for frequency above 1 kHz, the e l e c t r i c f i e l d i s e s s e n t i a l l y attenuated and n e g l i g i b l e f l u x outside the sheath. Thus, since character- i s t i c frequencies of l i g h t n i n g strokes exceeds 1 kHz, the above r e s u l t s i n d i c a t e that each phase of the cable i s decoupled from other phases as was shown previously i n Chapter 3. A f t e r the tangential current density f o r one medium i s obtained by equations (A.3) to (A.5), the tangential current density for another medium on the boundary to the f i r s t medium can be obtained by E l t = E 2 t °1 J l t = ~ 2 J 2 t Thusm the t o t a l current flowing i n d i f f e r e n t components of the cable system can be obtained by I = /JdA 109 BIBLIOGRAPHY 1. W. Diesendorf, 'Insulation coordination i n high voltage e l e c t r i c power systems', Butterworth Cp. London 1974. 2. M. 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(in Japanese)

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